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A cylindrical waveguide resonator for the investigation of ceramic breakdown at microwave frequencies Chute, Frederick Stephen 1964-12-14

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A CYLINDRICAL ¥AVEGUIDE RESONATOR FOR THE INVESTIGATION OP CERAMIC BREAKDOWN AT MICROFAVE FREQUENCIES by FREDERICK STEPHEN CHUTE 5,A.Sc.t U n i v e r s i t y of B r i t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g ¥e 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 UNIVERSITY OF BRITISH COLUMBIA December 1963 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 allowed without my w r i t t e n permission. Department of ^£^<^-T#/C# &/L/<S-/fl/££fi> //V<$— The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date ABSTRACT The design and c o n s t r u c t i o n of a microwave c a v i t y , to be used i n t e s t i n g the e l e c t r i c a l breakdown p r o p e r t i e s of t i t a n i a ceramic* i s d e s c r i b e d . The c a v i t y i s formed from c y l i n d r i c a l waveguide with a d i s c of t i t  a n i a c e n t r a l l y l o c a t e d i n the c a v i t y . Dimensions are so chosen t h a t resonance i s obtained i n the H , mode at o l 3000 mcs« The c a v i t y f i e l d s are d e r i v e d and from them the t h e o r e t i c a l c a v i t y Q i s found* Experimental values of Q are obtained from low power impedance measurements. Bethe's smal l - h o l e c o u p l i n g theory i s a p p l i e d to the design of an i r i s to couple power to the c a v i t y from the waveguide system. The dynamics of e l e c t r o n s i n t h i s t e s t c a v i t y are i n v e s t i g a t e d . Time-averaged t r a j e c t o r i e s and upper l i m i t s to e l e c t r o n energies are d e r i v e d from the Hamiltonian of the motion* V i t h t h i s i n f o r m a t i o n , the s i g n i f i c a n c e of e l e c t r o n bombardment and m u l t i p a c t o r e f f e c t s , i n the ceramic breakdown, i s d i s c u s s e d . The c a v i t y i s used to t e s t g l a z e d and unglazed t i t a n i a ceramic specimens at high f i e l d s t r e n g t h s , u s i n g a 2Mw magnetron. In a d d i t i o n , the e f f e c t i v e n e s s of an aluminum-titania s e a l on the d i s c edge i s i n v e s t i g a t e d . I t i s found t h a t * i f s u i t a b l e p r e c a u t i o n s are taken, the surface breakdown strengt h of t i t a n i a i s i n Kv excess of 50 — when subjected to a p u r e l y t a n g e n t i a l E f i e l d . ACKNOWLEDGEMENT The author wishes to express h i s thanks to Dr* G. Bo Walker, the s u p e r v i s o r of t h i s p r o j e c t , f o r many hours of v a l u a b l e d i s c u s s i o n and f o r h i s u n f a i l i n g encouragement• To Dr. R. Hayes, Mr» R« Sto c k w e l l , Dr„ C. E n g l e f i e l d , Mr. C, R, James* and many other c o l l e a g u e s , the author extends h i s s i n c e r e a p p r e c i a t i o n . Acknowledgement i s g r a t e f u l l y given to the Nat i o n a l Research C o u n c i l f o r the a s s i s t a n c e r e c e i v e d from a Bursary awarded i n 1962* a studentship awarded i n 1963 and funds f o r re s e a r c h through the Departmental Block Term Grant. TABLE OP CONTENTS Page L i s t of I l l u s t r a t i o n s •..».«».«*«.«..•«•»«.• v i Acknowledgement ..•....«.»».«.«.•.•«....»•»• v i i 1« I n t r OdUC 11 On ....»»» .^cra...a«»ooooo.oo 1. l o l B r i e f Summary of Breakdown at Ceramic Surfaces . «« « . » a <> » • . • « »• • 1 1.2 I n v e s t i g a t i o n s Leading To Present 1.3 Object of Present Work ............ 4 2. C a v i t y Design and T e s t i n g .............. 5 2.1 IntrO dU C t i o n ..;...»«»».«.«. .o. ...... 5 2.2 C a v i t y F i e l d s — Matching C o n d i t i o n and C a v i t y Parameters 9 2.3 Design of Coupling Aperature ...... 14 2.4 T h e o r e t i c a l Unloaded Q of Disc Loaded 2.5 A Method of Q Me asurement ........ 23 2.6 Low Power Test R e s u l t s ............ 28 3» Ceramic Breakdown In the S—Band C a v i t y . 35 3*1 I n t r o d u c t i o n ..««».a................ 35 3.2 Breakdown Tests I and I I - Unglazed Di.SC • * « » » * e * o * 4 # « « 9 * « < » * » o * 0 o 0 6 » « » 0 33 3.3 Breakdown Test I I I - Glazed Disc ... 41 3.4 Breakdown Test V ~ Glazed Disc With Thin Aluminum Edge Coating ........ 42 3.5 Breakdown Test V - Glazed Disc with Improved Aluminum Edge Coating .... 44 3.6 D i s c u s s i o n of R e s u l t s ............ • 46 i v Page 4> E l e c t r o n Motion In The S*-Band C a v i t y .... 51 401 I n t r o d u c t i o n . • « . . . . . . . . s o . « 51 4 02 Approximate Numerical S o l u t i o n by Method of Bunge—Kutta .............. 52 4.3 P o t e n t i a l Expressions f o r C a v i t y Fx elds . « . . . . « . . « « . « . . . . o ' * « . « o . . . . . . . 54 4.4 Hamiltonian and the Equations of Motion .... ...«•.»... «... 55 4.5 E l e c t r o n Emitted at Instant of Maxi mum E^-Time-Average Approximation . . 57 4.6 E l e c t r o n Emitted at Ins t a n t of M i n i  mum E^-Time-Average Approximation 64 4.7 E s t i m a t i o n of E r r o r Introduced by the Assumption of Time^Averaged Values 68 4.8 Lorentz Time-Average Force ......... 69 4*9 Multxpactor .«..«.»....«.e»..««..o.« 72 4.10 Summary ...................oooeooo.. 73 5» Conelusions .««.««.*«»..*..««.•«.• ........ 75 Appendix 1, Lorentz Time—Averaged Force .... 77 Beferences ..................a............... 80 v LIST OF ILLUSTRATIONS Figu r e Page 2*1 Cross S e c t i o n of S-Band Cavity- a a e e a a e © a 7 2*2 C a v i t y Layout ••••••««•••••«»a o a a a a a a a a a 8 2*3 C a v i t y Assembly *•»•••••«••<>». aa* a a a a a a a . 8 2*4 C a v i t y Schematic «•*••««•«««»• a a a a a o a o a • 11 2*5 Coupling C o n f i g u r a t i o n >.-*•••« a a a a a a a a. a a 15 2*6 Lumped Element E q u i v a l e n t ; C i r c u i t of 24 Coupling • a a a a a- • » • • •'*•• • s « • s a o a a a a a a a a a a 2*7 VS¥R C h a r a c t e r i s t i c ««««•«a».o a BJD a a a a a a a 30 2.8 VS¥R C h a r a c t e r i s t i c a o a a ft a a a a a a 31 3.1 High Power System *»».«••*. e . a a a a « 9 * o a » a 36 3*2 C a v i t y Pumping Scheme •*•»•... a a a a a a * a a a 37 3*3 Ceramic Breakdown - Test I I a a 40 3.4 Ring D i s c o l o u r a t i o n Test II a a a 9 0 9 a a a a 40 3.5 Ceramic Breakdown — Test IV •. 45 3*6 45 4.1 T y p i c a l T r a j e c t o r i e s «••••«. a a a a a a a a a a a a 54 4.2 (kr) Approximation *••» ,. a . a a a a a ft a a a a 59 4>3 Time-Averaged E l e c t r o n Motion a a a o a a a a a a 71 v i 1 A CYLINDRICAL WAVEGUIDE RESONATOR FOR THE INVESTIGATION OF CERAMIC BREAKDOWN AT MICROWAVE FREQUENCIES 1. INTRODUCTION 1.1 B r i e f Summary of Breakdown at Ceramic Surfaces. Microwave tubes and more r e c e n t l y d i e l e c t r i c loaded slow wave s t r u c t u r e s make extensive use of d i e l e c t r i c m a t e r i a l s . In many a p p l i c a t i o n s i t has been found that breakdown of these d i e l e c t r i c s has been the l i m i t i n g f a c t o r on the power handling c a p a b i l i t i e s of the d e v i c e s . F a i l u r e of ceramics at microwave frequencies has been under i n v e s t i g a t i o n by groups at 1 2 3 Stanford U n i v e r s i t y ' , Sperry Gyroscope Co. , E l e c t r i c 4 5 and Musical I n d u s t r i e s L t d . « E i t e l - M c C u l l o u g h Inc. , and a number of other c e n t r e s * Ceramics f a i l i n a v a r i e t y of ways. The most common types of f a i l u r e are % (a) A puncture through the m a t e r i a l . (b) S t r u c t u r a l damage r e s u l t i n g from thermal shock. (c) Surface sparking* In a l l the i n v e s t i g a t i o n s a s s o c i a t e d with the present study, type (c) i s the p r i n c i p a l source of breakdown. This form of ceramic f a i l u r e w i l l be r e f e r r e d to as surface breakdown and i t appears to be c r i t i c a l l y 2 dependent on such f a c t o r s as surface c o n d i t i o n , contamination* m e t a l - d i e l e c t r i c contacts and m a t e r i a l composition, 1,2 I n v e s t i g a t i o n s Leading to Present Study, The present experiments were o r i g i n a t e d to i n v e s t i g a t e d i e l e c t r i c f a i l u r e s observed during work on d i e l e c t r i c loaded p e r i o d i c s t r u c t u r e s at Queen Mary C o l l e g e , U n i v e r s i t y of London. The breakdown r e s u l t e d i n r a d i a l t r a c k s of a t r e e — l i k e p a t t e r n on the d i e l e c t r i c s u r f a c e . In a d d i t i o n there was a l i g h t brown d i s c o l o r a t i o n on the i n t e r i o r metal w a l l s and on the surfaces of the d i e l e c t r i c d i s c s . These l o a d i n g d i s c s were composed of a s i n t e r e d ceramic c o n s i s t i n g of tita.nium d i o x i d e , with small amounts of a d d i t i o n a l compounds to improve s t a b i l i t y * Samples were obtained from Dr. E, Schaefer, The Laboratory, 30 Thompson's Lane, Cambridge, England* I n i t i a l experimentation showed that the breakdown strength of t i t a n i a can be g r e a t l y i n c r e a s e d by c o a t i n g 6 the ceramic with a l e a d borate glaze • This glaze was obtained from Blythe Colour "Works L t d , , C r e s s w e l l , Stoke- on—Trent* as a suspension of f i n e p a r t i c l e s i n water. I t was sprayed on the ceramic and heated to 425°C to provide the g l a z e d s u r f a c e . F u r t h e r i n v e s t i g a t i o n was c a r r i e d out at the 3 U n i v e r s i t y of B r i t i s h C o l u m b i a ' y ' o p , i U , The o b j e c t vas to determine what p r o p e r t i e s of the glaze were r e s p o n s i b l e f o r the improvement i n breakdown s t r e n g t h . To t h i s end* t e s t s were made of the surface c o n d u c t i v i t y and secondary emission of both the t i t a n i a ceramic and the l e a d borate g l a z e * As a r e s u l t of these t e s t s * i t was concluded t h a t d i f f e r e n c e s i n surface c o n d u c t i v i t i e s and secondary emission were not r e s p o n s i b l e f o r the improved breakdown p r o p e r t i e s . A d d i t i o n a l c l a r i f i c a t i o n of the breakdown process was obtained from experiments w i t h s t a t i c and dynamic f i e l d s * P l a i n t i t a n i a was shown to s u f f e r a permanent damage at the i n s t a n t sparking occurs. Breakdown t r a c k s , c o n s i s t i n g of a lower semiconducting oxide, are formed by the intense l o c a l h e a t i n g a s s o c i a t e d with the e l e c t r i c a l discharges and* once formed, these t r a c k s permanently des t r o y the i n s u l a t i o n p r o p e r t i e s of the s u r f a c e . A p p l i c a t i o n of the t h i n glaze c o a t i n g appears to form a p r o t e c t i v e l a y e r on the t i t a n i a surface that prevents the formation of the semiconducting oxide. As a r e s u l t , i n i t i a l random s p a r k i n g , occasioned by surface contamination, i s not permanently damaging. A spark c o n d i t i o n i n g process takes place and the breakdown t h r e s h o l d then r i s e s to a much higher v a l u e . In the m a j o r i t y of cases^ the breakdown pat t e r n s o r i g i n a t e d at a metal-ceramic c o n t a c t . I n s p e c t i o n of the 4 contact u s u a l l y r e v e a l e d t h a t the metal and ceramic d i d not f i t evenly at the p o i n t * It was concluded that the breakdown was i n i t i a t e d by f i e l d emission i n the gaps formed at the improper metal—ceramic c o n t a c t . Attempts to e l i m i n a t e t h i s edge breakdown by s i l v e r i n g the ceramic edge f a i l e d at microwave f r e q u e n c i e s but had some success at d*c« This f a i l u r e of the m e t a l — t i t a n i a s e a l prevented adequate t e s t i n g of the surface breakdown s t r e n g t h of the l e a d borate g l a z e d d i s c s at microwave f r e q u e n c i e s , 1*3 Object of Present Work* The problems a s s o c i a t e d w i t h the edge breakdown prompted the i n v e s t i g a t i o n of a c y l i n d r i c a l S-Band c a v i t y o p e r a t i n g i n a mode w i t h zero E f i e l d at the outer circumference, f o r the purpose of e l i m i n a t i n g metal- ceramic f a i l u r e s * Such a mode i s the i n a c y l i n d r i c a l waveguide, since the E f i e l d c o n s i s t s only of an Eg component. The remainder of t h i s t h e s i s i s concerned with the design and alignment of the c a v i t y and the f u r t h e r t e s t i n g of glazed and unglazed t i t a n i a d i s c s * In a d d i t i o n , an assessment i s made Of the pa r t played by ene r g e t i c f r e e e l e c t r o n s i n the breakdown i n t h i s c a v i t y . The breakdown experiments w i l l i n c l u d e t e s t s on the e f f e c t i v e n e s s of an aluminum—titania s e a l i n preference to the s i l v e r c o a t i n g t h a t f a i l e d i n previous experimental work* 2. CAVITY DESIGN AND TESTING 2*1 I n t r o d u c t i o n . The f i r s t experimental c a v i t y c o n s i s t e d of a c y l i n d r i c a l waveguide c a v i t y with a t i t a n i a d i s c c e n t r a l l y l o c a t e d . The ceramic was h e l d i n a copper r i n g shrunk onto the outer edge* Dimensions were chosen so t h a t the c a v i t y was resonant i n the H , mode at J o l 3000 mcs. The c a v i t y diameter was d i c t a t e d by the diameter of a v a i l a b l e d i e l e c t r i c d i s c s to be 7.70 cm, and hence, the H , mode was c u t o f f i n the a i r s e c t i o n 1 o l of the c a v i t y . Resonance was obtained with a d i s c t h i c k ness of 0.582 cm* The c a v i t y was coupled, through one end-^wall, to the narrow side of the waveguide run. The side c o u p l i n g proved v e r y u n s a t i s f a c t o r y and t h i s small c a v i t y was abandoned t e m p o r a r i l y * The problem of c o u p l i n g power to a c a v i t y i n the ^ o l m ° ^ e w a s f u r t h e r i n v e s t i g a t e d u s i n g a 5 i n c h diamete c a v i t y w i t h no ceramic d i s c * In t h i s case power was coupled to the c a v i t y , from the end of the waveguide run* through a c i r c u l a r i r i s i n one end-wall. This c o u p l i n g end-wall was co n s t r u c t e d as a' removable p l a t e thus f a c i l i t a t i n g the r a p i d changing of i r i s dimensions. The c o u p l i n g was extremely e r r a t i c and repeatable r e s u l t could not be obtained. Since s l i g h t v a r i a t i o n s i n pressure on the b o l t s supporting the c o u p l i n g p l a t e 6 caused l a r g e f l u c t u a t i o n s i n the VSWR i n the input waveguide, a mechanically r i g i d c o u p l i n g p l a t e was con s t r u c t e d . This m o d i f i c a t i o n was accomplished by br a z i n g the brass c o u p l i n g p l a t e over the end of the waveguide. C o n s i s t e n t measurements were then obtained. Another problem arose due to the simultaneous e x c i t a t i o n of the E,, mode as w e l l as the d e s i r e d H , 11 o l mode* The i n t r o d u c t i o n of a probe i n t o the c a v i t y a f f e c t e d the c o u p l i n g , apparently as the r e s u l t of mode c o n v e r t i n g * A non-conducting washer was f i t t e d between the c y l i n d r i c a l w a l l s and the end p l a t e s of the c a v i t y * thus damping the E^^ mode* Using the s o l i d c o u p l i n g p l a t e and the E ^ mode f i l t e r * the input VSWR was reduced below 3,0 at resonance. At t h i s p o i n t , because of the encouraging r e s u l t s with the l a r g e r c a v i t y , end c o u p l i n g was adapted to the small c u t o f f c a v i t y * In t h i s case the E^^ mode i s not e x c i t e d at the same frequency as the because of the presence of the lo a d i n g d i s c * The frequency s e p a r a t i o n was s u f f i c i e n t to e l i m i n a t e the need f o r the E ^ mode f i l t e r . The c o u p l i n g was an immediate success, with values of VSWR l e s s than 2,5* at resonance* F i g u r e s 2*1* 2,2, and 2*3 i l l u s t r a t e the p h y s i c a l layout of t h i s c a v i t y . P i g , 2.1 C r o s s S e c t i o n o f S - B a n d C a v i t y P i g . 2*3 C a v i t y Assembly 9 The remainder of t h i s s e c t i o n i s devoted to the design and low power t e s t i n g of t h i s c u t o f f c a v i t y . The p h y s i c a l dimensions and t h e o r e t i c a l unloaded Q (Q 0) are c a l c u l a t e d . A l s o , the c o u p l i n g hole s i z e , e x p e r i m e n t a l l y obtained* i s compared with the diameter given by Bethe's small—hole c o u p l i n g theory* 2*2 C a v i t y F i e l d s - Matching C o n d i t i o n and C a v i t y Parameters* The c a v i t y i s designed to operate i n the mode at 3000 mcs, with the ceramic d i s c c e n t r a l l y l o c a t e d . The f i e l d s are so chosen t h a t the centre of the d i s c and both end w a l l s are nodal planes of the tra n s v e r s e E f i e l d . The waveguide f i e l d i n c y l i n d r i c a l coordinates • .11 1SJ H = H J (krOe^^ 2 Z 0 0 H R = | RoJ1(tec)ei(*±-Xz *..2.1 E = - M . H J 1 ( k r ) e J f l , t - i r z © k o 1 N assuming p e r f e c t l y conducting metal w a l l s and l o s s l e s s d i e l e c t r i c , "fl"' i s the complex propagation constant and J2 = k 2 - co2|iE ...2.2 10 The addition of two t r a v e l l i n g waves of the form given by equations 2*1 yields the f i e l d for an empty cavity* Thus with z = o a nodal plane, the cavity f i e l d becomes: H = -H,J ( k r O s i n h T z ' e ^ z 1 o v - u H r = ^  H ^ t k x J c o s h T z - e ^ " 6 ...2.3 The r a d i u s of the c a v i t y i s f i x e d , by the diameter .of a v a i l a b l e ceramic d i s c s * at a = 3.85 cm. In order to s a t i s f y the boundary c o n d i t i o n s * must be zero at r = a* For the H , mode t h i s c o n s t r a i n t r e q u i r e s that o l ka i s the f i r s t zero of J ^ ( k r ) * Hence, ka = 3.83 and k = 99*5. Thus, from equation 2*2* "£ = jB = j598 i n the d i e l e c t r i c r e g i o n and "jf = a = 77*2 i n the a i r r e g i o n . The f i e l d s outside the d i e l e c t r i c d i s c a r e , t h e r e f o r e , expressed i n terms of h y p e r b o l i c f u n c t i o n s while those w i t h i n the d i s c are represented by t r i g o n o m e t r i c f u n c t i o n s . The f i e l d p a t t e r n i n the d i e l e c t r i c loaded c a v i t y can be obtained by c o n s i d e r i n g standing waves, of the ! form given by equations 2*3* to e x i s t i n both the a i r and d i e l e c t r i c p o r t i o n s of the c a v i t y . By matching f i e l d components at the d i e l e c t r i c — a i r i n t e r f a c e , i t i s p o s s i b l e to r e l a t e a and 8 to the p h y s i c a l dimensions of the c a v i t y * To s i m p l i f y the e x p r e s s i o n s , d i f f e r e n t z = o planes are chosen i n each s e c t i o n . Figure 2.4 shows a c r o s s — s e c t i o n a l view of the c a v i t y with these o r i g i i i s marked* F i g * 2.4 C a v i t y Schematic The appropriate f i e l d s ares (a) In the a i r s e c t i o n H z = -H, J ( k r ) s i n h otz-e 1 o H r = ^ H^J^(kr)cosh az»e^ W^ ...2. EQ = ^ ± H - j J ^ k r J s i n h a z « e j W t (b) In the d i e l e c t r i c s e c t i o n H z = H 2 Jo^ k r^ s i n^T e : J f l ) t H r = =| H 2 J 1 ( k r ) c o s S / T j » e ; j W t ...2. E Q = - ^ 2 - H ^ k r J s i n B ^ e ^ At the d i e l e c t r i c - a i r i n t e r f a c e the f o l l o w i n g boundary c o n d i t i o n s must be s a t i s f i e d : ^ t a n g e n t - a i r ^ t a n g e n t — d i e l e c t r i c ^ ) ^ t a n g e n t — a i r ^ t a n g e n t - d i e l e c t r i c ••• 2' normal-air n o r m a l - d i e l e c t r i c 13 Thus H^u^sinhaL = I^pi^sinBd H^acoshpcL = — ^ S c o s B d D i v i d i n g these two equations y i e l d s the matching c o n d i t i o n s -^2 a tanhocL»cos8d = • 3- ,.,,2.7 »l $ I f d i s to give a p r a c t i c a l d i s c t h i c k n e s s , (3d must l i e between ^ and 7 1 * To make the most e f f i c i e n t use of a v a i l a b l e power i t i s d e s i r a b l e to have a maximum E Q f i e l d at the d i s c s u r f a c e * This c o n d i t i o n i m p l i e s that IT 8d must approximately equal # but then ocL must approach i n f i n i t y to s a t i s f y equation 2.7., The l a r g e r aL becomes,the more d i f f i c u l t i t i s to couple power to the c a v i t y . A s a t i s f a c t o r y compromise i s aL = 1. Hence, from the matching c o n d i t i o n * the c a v i t y parameters ares C a v i t y Length (2L + 2d) **•»*••*•**••*... 0.0317 meter C a v i t y Radius (a) ..... ...*............. * 0.0385 meter Disc Thickness (2d) .......».•*•*•....... 0.00582 meter a . . . . . . . . . . . * o . . . ........ 77.2 nepers meter 8 .........**»*.••*«...• 598 radians meter C h a r a c t e r i s t i c Number (k) »<,..•><>........ 99.5 R e l a t i v e P e r m i t t i v i t y ( e r ) of t i t a n i a d i s c .,...».*«**•*«•*»«««.».. 93 14 2*3 Design of Coupling Aperture* Power i s coupled to the c a v i t y from the end of the waveguide through a c i r c u l a r i r i s i n one of the c a v i t y end w a l l s . F i g u r e 2*5 i l l u s t r a t e s the f a s h i o n i n which c o u p l i n g i s obtained* 12 F. Shnurer * u s i n g the small hole c o u p l i n g theory 13 developed by Bethe , has r e l a t e d the c o u p l i n g hole diameter to the loaded Q (Qjj) °^ a c a v i t y . Shnurer proceeds by s e t t i n g n _ <•> x s t o r e d energy  L S where S i s the power l o s t through a c o u p l i n g hole i n a c a v i t y coupled to a waveguide system. I t i s f e l t t h a t t h i s expression i s i n e r r o r i n n e g l e c t i n g the w a l l l o s s of the c a v i t y . Consider a c a v i t y coupled to a waveguide system by a l o s s l e s s window. I t can be shown t h a t d e f i n e s the r a t e at which t h i s c a v i t y l o s e s energy i n the absence of e x c i t a t i o n * This energy l o s s i s comprised of the w a l l l o s s p l u s the energy r a d i a t e d through the c o u p l i n g h o l e , i n other words + S. For u n i t y c o u p l i n g P = S and the e x p r e s s i o n f o r should be. n - x s t o r e d energy _ <o x s t o r e d energy  y L ~ •+ S ~ 2S -W 15 F i g * 2»5 Coupling C o n f i g u r a t i o n 16 In the f o l l o w i n g c a l c u l a t i o n s the m o d i f i e d value of w i l l be used but the method w i l l be the same as o u t l i n e d by Shnurer* Bethe has shown t h a t the power l o s t , through a window i n a c a v i t y , i n t o the end of a waveguide can be expressed a s j k 2 S = £ Sa PE E + jM,H -H » + jM 0H H on an J 1 or af d 2 om am 2 • . . 2«8 where P, M^, M^* are dependent on window geometry and are known as p o l a r i z a b i l i t i e s * The f i e l d s with s u b s c r i p t "o" are c a v i t y f i e l d s , those with s u b s c r i p t "a" are waveguide f i e l d s * The s u b s c r i p t s f * m, n, r e f e r to the axes of the window. Sa i s a n o r m a l i z i n g constant.. Here 3 3 i P = d g f M i = M 2 = ^ a , where *|==,/f , k Q = & , and d i s the i r i s diameter* The dominant mode f i e l d s i n the r e c t a n g u l a r guide .14 are i 3 ( a t - 0- z) TT TT TtX. . - 0 H - ELcos -— *e z 2 c i2c " itx j ( < 0 t ~ P o z ) H = l £ C H i n 2L2L * e 0 ...2,9 x A 2 c g •o j (»t - S z) y X n ' 2 c The Coupling f i e l d s i n v o l v e d i n t h i s case are those f i e l d s 17 i n the c a v i t y and the waveguide at z = o, x r = 1.925 cm. Hence ~ ~2 t and S = Sa jM,H H J 1 r 3 H r = I H i J l ( k r ) H - '1 2 c H x X n 2 Therefore S = ^ o_ Sa ..2.10 But from Bethe Sa = J Real (E^, x H T*)ds where, i n t h i s case, E™ and H„, are the tr a n s v e r s e waveguide f i e l d s . b c - " ] H 2 H 2 j j s i n 2 = a*dy 0 ' ^ 0 0 4c^ H 2 X X 2 o g 3 2 2c b ^ o g ..2.11 Thus, s u b s t i t u t i n g J^(k.1*925).-= (.5807)' 0.440H2d/>| " T T ~ c b o g • • o 2 • 12 18 and n - tt x e n e r g y stored  W L _ 2S « » a 2 0 X 3 The stored energy i s c a l c u l a t e d i n s e c t i o n 2.4 and i s given as 81t3ftp x 1 0 " 1 5 H 2 j o u l e s . Hence QT = 0.348A, cb g 2&\ and ,/ , 0.348A cb d 6 = — - S — ...2.14 For the c r i t i c a l l y cdupled case = —— , Q q i s given i n s e c t i o n 2.4 as 2850. Therefore J T = 1425 and X = L g 1 - r = 13.9 cm also c =7.2 cm and b = 3*4 cm. A | = 377 . Hence and -,6 ^  , , , rs-10 6 d «; 1*1 x 10 m d 2.18 cm Ex p e r i m e n t a l l y i t was found t h a t an input standing wave r a t i o of l e s s than 2*0 could be obtained with a hole diameter of 2.60 cm* Thus* the t h e o r e t i c a l diameter d e r i v e d above i s l e s s than 15 percent i n e r r o r . The d e v i a t i o n can be a t t r i b u t e d p a r t l y to the f a c t that the maximum Q was used i n the c a l c u l a t i o n s and not the a c t u a l / 0 19 Q q which i s much lower. Using the a c t u a l Qo w i l l l e a d to a l a r g e r diameter. Als o i t must be remembered that Bethe made the approximation t h a t d ^< X Q. The useable accuracy achieved i n c a l c u l a t i n g coupling,holes i n the above f a s h i o n , however, i n d i c a t e s that Bethe's s m a l l - hole theory provides a s a t i s f a c t o r y approximation f o r XQ holes of the order of — r i n diameter. This same 4 obse r v a t i o n was made by Shnurer i n h i s work on aperture- coupled f i l t e r s . 2*4 T h e o r e t i c a l Unloaded Q of Disc Loaded Resonator, The unloaded Q of a c a v i t y i s given by Q _ toxstored energy *o — mean power d i s s i p a t e d ,2.15 The s t o r e d energy, e g , i s * e s = 2 v o l 2 e |E | +2^|H| dv and since f o r the l o s s l e s s c a v i t y V r HHI: | E | E | 2 dv = 'dv e E dv v o l The complete e c o n s i s t s of two p o r t i o n s ; the energy s t o r e d s 20 i n the d i e l e c t r i c r e g i o n and the energy s t o r e d i n the a i r 1 / space* ,/ In the a i r space: / a L 2 2 eg'' = 2 x ^ 2 ^ 2 ^ 1 ^ 2r J/j ( k r ) d r \ sinh 2oczdz = 2 x ^ | » 2 ^ i H i * | - Jo(ka) #5.26 x 10~ 3 = 1984 x 1 0 ~ 1 5 H 2 j o u l e s ...2.16 In''the d i e l e c t r i c r e g i o n : / /rt2 2 T2 d ' 1 • ^2 „ TT2 2 Jc 'e . ^ T t H^a^lka) J sin 2B^d/*j e s - 2 X 2 ,2 - ^ ' ^ ^ o e = 56,000 x 10"" 1 5H 2 s ' 2 and from equation 2.6, H-^  = 0«84H2* Therefore e = 79,400 x 1 0 " 1 5 H 2 j o u l e s ...2.17 s 1 A good approximation of the w a l l l o s s i n the c a v i t y can be made by u s i n g the value of H^ , at the w a l l and the wave impedance of the w a l l * 21 T h e n P - I L 2 A g H T | 2 d s •where A i s t h e s k i n d e p t h a n d i s g i v e n a s A = ( ^ ^ ) g i s t h e c o n d u c t i v i t y o f t h e w a l l * T h e p o w e r l o s t t o t h e b r a s s e n d w a l l s i s : ' a 2n 2 P l = I i i I I ? H ^ ( k r ) a s x 2 k o o ° L J L _ H 2 - ^ . J 2 ( k a ) x 2 A g k ^ 1 * 0 1 2 . 7 2 x 1 0 * * 6 H 2 w a t t s . . . 2 . 1 8 T h e p o w e r l o s t t o t h e c o p p e r s i d e w a l l s i n t h e d i e l e c t r i c p o r t i o n i s : d P 2 = f | H 2 J o ( k a ) ^ s i n 2 B / » | . d r f j x 2 1 . 2 7 x 1 ( T 6 H 2 w a t t s . . . 2 . 1 9 T h e p o w e r l o s t t o t h e s i d e w a l l s i n t h e a i r r e g i o n i s : ^ P3 = f | H l J o ^ k a ^ s i n h 2 a z d z x 2 o = 2 . 9 6 x 1 0 ~ 6 H 2 w a t t s . . . 2 . 2 0 15 The power l o s t to the d i e l e c t r i c i s given as s PD = 1 j v g | E ! 2 < i v ...2.21 A' V EI 2dv, tan6 = 0.00035 f o r t i t a n i a . 2 P D = e2tanS<o3(j,2,ixH2J2(ka)—2 x 1.59 x 10~ 3 x 2 2k = 521 x 10~*^H2 watts ...2.22 The r e s u l t s f o r the l o s s e s a re: —6 2 end w a l l s •.«....«.«»«».»« 12.72 x 10 H-^  6 2 side w a l l s .... .......«««• 1.27 x 10 —6 2 ......««».««»*». 2.96 x 10 —6 2 d i e l e c t r i c «.......*«...«. 521.0 :.:x 10 538 x 10~ 6H 2 watts The r e s u l t s f o r the st o r e d energy ares -15 2 d i e l e c t r i c r e g i o n 1984 x 10 a i r r e g i o n 79400 x 1 0 ~ 1 5 H 2 81400 x 10~ 1 5H^ j o u l e s 23 Hence from equation 2*15 the unloaded Q i s . 6.28 x 10 9 x 3 x 81400 x 1 0 ~ 1 5 H 2 e0 = 538 x 10""6H2 C- = 2850 • o 2.5 A Method of Q Measurement* o The Q of a c a v i t y can be obtained from a p l o t of VSWB. i n the input waveguide, against frequency about the resonance p o i n t , using a method described by G i n z t o n 1 ^ . A knowledge of the coupling i s required and t h i s i s expressed i n terms of a coupling c o e f f i c i e n t . and at the h a l f power frequencies* thus enabling QQ to be determined d i r e c t l y from standing wave r a t i o data. resonator equivalent c i r c u i t shown i n Figure 2.6. ZQ i s the waveguide c h a r a c t e r i s t i c impedance and i s the s e l f inductance of the coupling hole* G* L, R are c a v i t y s parameters and M represents the coupling between the waveguide and the c a v i t y * Consider the impedance coupled i n s e r i e s with the c a v i t y parameters. t u r n , i s e a s i l y r e l a t e d to the VSWR at resonance Y i s defined i n terms of/ the parameters of the 24 which may be written as where and hence • • •2•23 The t o t a l impedance terminated i s given ass with which the l i n e i s 25 where Z L = J W L 1 + Z Z = 2 2 ttC-1 R J Y J * 0 CO ...2.24 The i n t e r p r e t a t i o n of the impedance expressions can he aided by choosing s p e c i a l r e f e r e n c e planes, along the waveguide run* at which the term r e p r e s e n t i n g the s e l f — r e a c t a n c e of the c o u p l i n g system d i s a p p e a r s . In g e n e r a l , the impedance of these planes, i n terms of the l o a d impedance f i s : Z L + j Z Q t a n 8 o l 'o Z + i Z T tanB 1 o J L r o ...2.25 where "1" i s the d i s t a n c e from the l o a d to the reference plane* To l o c a t e such a r e f e r e n c e plane, detune the c a v i t y and l o c a t e a v o l t a g e node along the waveguide. This p o s i t i o n is- c a l l e d the detuned-short. 26 At the detuned-short. with the c a v i t y detuned, the impedance i s zero and from equation 2.24 Z-^  = jttL^. S u b s t i t u t i n g these values i n t o equation 2.25 y i e l d s : - c o L 1 tan 8 1 = „ r o Z o Thus* the impedance at the detuned short i n the v i c i n i t y of resonance can be w r i t t e n as: 1 .+ CO 2 Z o 2 o ...2.26 'o 1 + J2Q Q(M - M 0) • ••2«2T At c e r t a i n f r e q u e n c i e s = o 1 + j l hence, 2Q o(M 1 - M ) = 1 or 2Q Q(M 2 -M Q) = -1, Therefore CO M - M U l n2 ( c o r co o ) - ( c o 2 - <oo) con CO, CO — CO 1 2 CO CO - CO ~ f — f 1 2 1 2 27 Thus* f-j^ and are the h a l f power f r e q u e n c i e s , from equation 2.26, At these v a l u e s of frequency the impedance at the detuned short i s * z c i f Z^ = 1 + j l ••..2.28 The corresponding value of VSWR at these 17 frequencies can he obtained from i Z„ + Z I .+ Z„ - Z I e 1 c 0' 1 c o i 0 O Q S . = |Z +.Z I H Z - Z-' . . . 2 . 2 V where S i s the VSVR. Hence* the value of S at the h a l f power fr e q u e n c i e s i s . 2 2 y y can be found i n terms of the standing wave at resonance f o r two d i s t i n c t cases, when V > 1 and when ^P< 1. At Z resonance ^— = ^ and from 2*29* ^> y 1 (overcoupled) •».»•• ^ = SQ ^ < 1 (undercoupled) ..«•* = 4— o 28 where S i s the VSWR at resonance* o \ ., \ ! The procedure f o r f i n d i n g Q i s as f o l l o w s : \ i {a) P l o t VSWR aga i n s t frequency about resonance, (b) Locate detuned sho r t and b r i n g c a v i t y c a r e - f u l l to resonance* (c) Explore the vo l t a g e d i s t r i b u t i o n around the detuned short p o s i t i o n u s i n g a s l o t t e d l i n e s e c t i o n . A v o l t a g e maximum means ^  = S Q A v o l t a g e minimum means ^ = (d) C a l c u l a t e S i and from the graph (a) f i n d 2 f o f, and f - . Then Q = -r- 7 — . 1 2 *o f 1 - f 2 2*6 Low Power Test R e s u l t s * The low power t e s t equipment was used to measure the Q o of the c a v i t y and to adj u s t the resonance as near as p o s s i b l e to 2998 mcs, the f i x e d frequency of the high power magnetron* Some s l i g h t c o n t r o l on the resonant frequency was needed to enable the frequency to be set to 2998 mcs each time the c a v i t y was assembled. These small v a r i a t i o n s i n the resonance p o i n t were obtained by p l a c i n g t h i n aluminum washers between the ceramic d i s c and the metal side w a l l s . The washers e f f e c t i v e l y lengthened the c a v i t y , thereby lowering the resonant frequency* 29 A small amount of power was coupled out of the c a v i t y by an H f i e l d probe and monitored. The resonant frequency was taken as the frequency at which t h i s out put power was a maximum. Each time the c a v i t y was assembled, p r i o r to a high power t e s t , the resonant frequency was set at 2998 mcs and a p l o t of VSWR aga i n s t frequency was taken on the low power bench. T y p i c a l graphs of VSWR versus frequency under a number of d i f f e r i n g c o n d i t i o n s are shown i n f i g u r e s 2.7, and 2*8. Figure 2.7 The d i s c i n t h i s case was glazed on one side with l e a d borate and the d i s c edge was aluminized u s i n g a vacuum evaporation technique* Here, u s i n g the method of s e c t i o n 2.5: S = 2 o - ~ (undercoupled) Si = 4.27 Af. = 2.56 Mc/s f = 2998*4 Mc/s 30 3000 — i ~ i 2999 2998 Frequency (mc/s) — i 2997 31 / 6 32 and Qo = AT = 1 1 7 0 F i g u r e 2.8 The d i s c had a glaze c o a t i n g on one side but the edge was not alumin i z e d . In a d d i t i o n , the c a v i t y was made vacuum t i g h t u s i n g a c y l i n d r i c a l brass j a c k e t e n c l o s i n g the e n t i r e c a v i t y * Here: S = 1*44 o ^ = ^/l«44 (undercoupled) S i = 3*27 ' 2 Af = 1*57 Mc/s f = 2998*2 Mc/s o and 2 0 = A? " 1 9 0 0 These values of Q are c o n s i d e r a b l y below the o t h e o r e t i c a l maximum given i n s e c t i o n 2.4. A p o r t i o n of the d e v i a t i o n can be a t t r i b u t e d to surface i r r e g u l a r i t i e s and the approximate nature of both the c a l c u l a t i o n s of c a v i t y l o s s e s and the g r a p h i c a l method used. The major source of e r r o r , however, i s the ne g l e c t of l o s s assoc- i a t e d with the t h i n l e a d borate g l a z e . This l o s s can be taken i n t o account by the f o l l o w i n g approximation: 33 = i \ ttetanolE D ~ 2 where E| = ^ ^ ( k r j s i n h a z . Assume that the cavity f i e l d s are approximately constant over the glaze thickness* d* Therefore a Pp = nxoetano^d ^ |E| 2 rdr o Ttde e u tt tan$ 0 0 2 0 = r ° 2 x 1.18 2H 2*f-.J 2(ka) Luthra 9 has determined e and tancS for lead borate r glazed on t i t a n i a as c = 19*3» tan5 = 0*033* r Hence Pp = 81*3 x 10~ 6H 2 watts/b.001" The glaze thickness i s not known exactly, but i t l i e s between 0.001" and 0*005"* If for example d = Q.003", P D 244 x 10""6H2 watts, and Q * 18*85 x 10 9 x 81400 x 10" 1 5  0 244 + 538 Therefore Q = 1960 o This value of Q q agrees c l o s e l y with the experimental data* 35 3. CERAMIC BREAKDOWN IN THE S-BAND CAVITY 3,1 I n t r o d u c t i o n , Breakdown t e s t s were c a r r i e d out using a high power microwave system c o n s i s t i n g of a 10 cm magnetron and modulator u n i t . Power was d e l i v e r e d i n the form of 2JAS pulses at a 60 cps recurrence frequency, and was continuously v a r i a b l e from a few k i l o w a t t s to a peak of 2 Mw, Although the magnetron o s c i l l a t e d at a f i x e d frequency, 2998 mcs, i t was p o s s i b l e to tune i t over a three to f o u r megacycle range by means of a phase s h i f t e r mismatch u n i t and by v a r y i n g the magnetron magnet c u r r e n t , A thermistor bridge was used to measure the power, and the frequency was monitored by a c a v i t y wavemeter. Figure 3,1 shows the high power system. In order to handle the power r e q u i r e d f o r break down t e s t i n g i t was necessary to evacuate the waveguide system. The c a v i t y was made vacuum t i g h t by e n c l o s i n g i t i n an o v e r s i z e d c y l i n d r i c a l j a c k e t which was sealed by "0" r i n g s to the flanges t h a t form the c a v i t y end p l a t e s . The probe hole wfas c l o s e d by an> "0" r i n g and a small piece of g l a s s , and i t was p o s s i b l e to view the back chamber under vacuum c o n d i t i o n s . The c a v i t y was pumped through the c o u p l i n g hole and waveguide run, the back chamber being evacuated through holes i n the side w a l l s * Figure 3*2 i l l u s t r a t e s the pumping scheme. The F i g . 3 . 1 High Power System 37 system was evacuated by a 3 inch mercury d i f f u s i o n pump, / with a l i q u i d n i t r o g e n t r a p , backed by r o t a r y pump. "0" Ring ^-Vacuum Jacket // // // if >i n n - - - -< y// n u if u zzXl P i g . 3.2 C a v i t y Pumping Scheme The f o l l o w i n g procedure was used f o r a l l the high power t e s t s . The c a v i t y was assembled with the ceramic d i s c to be t e s t e d and was placed i n the vacuum j a c k e t . Low Power measurements were made of the unloaded Q and **o the resonant frequency was set as c l o s e l y as p o s s i b l e to 38 2998 mcso The c a v i t y was then b o l t e d to the hi g h power waveguide and the system was evacuated. The breakdown t e s t i n g was commenced only a f t e r the pressure l e v e l , as measured on an i o n guage s e v e r a l f e e t from the c a v i t y i t s e l f , was reduced belowj10"^ mm Hg. I Power was i n c r e a s e d i n small steps u n t i l e i t h e r there was evidence of breakdown or the maximum power l e v e l was reached,, Each i n c r e a s e i n power caused a sudden pressure r i s e , due to outgassing of the waveguide w a l l s . The o r i g i n a l pressure l e v e l was r a p i d l y r e s t o r e d by the d i f f u s i o n pump. At each step the magnetron frequency was v a r i e d over a 3 to 4 megacycle range to ensure that power was d e l i v e r e d to the c a v i t y at i t s resonant frequency. 3»2 Breakdown Test I and I I - Unglazed D i s c . The experimental procedure f o r these two t e s t s was e x a c t l y as o u t l i n e d i n the I n t r o d u c t i o n . In a d d i t i o n , a Geiger-Muller Counter was l o c a t e d near the c a v i t y f o r the purpose of monitoring any r a d i a t i o n generated by en e r g e t i c e l e c t r o n s as a r e s u l t of break down at the ceramic d i s c . The power was g r a d u a l l y i n c r e a s e d to a peak of 1*2 Mw, a l l o w i n g s u f f i c i e n t time f o r outgassing of the system. The Geiger Counter r e g i s t e r e d only the random r a d i a t i o n from the magnetron power supply and modulator 39 and there was no change i n count when a breakdown occurred. E x t e r n a l l y , the only s i g n of breakdown was the considerable i n c r e a s e i n the temperature of the c a v i t y . In both t e s t s d i s m a n t l i n g the c a v i t y - r e v e a l e d a large d i s c o l o u r e d area on the d i s c surface nearest the c o u p l i n g h o l e . This breakdown p a t t e r n appeared to be an extensive network of i n t e r l o c k i n g t r a c k s of reduced t i t a n i a . No other breakdown was evident any where on the d i s c . The reduced r e g i o n was c e n t r a l l y l o c a t e d and about 3 cm i n diameter. W i t h i n t h i s breakdown area the d i s c surface contained a number of p i t s , some, about 0.1 cm deep. The m a j o r i t y of the t r a c k s of reduced t i t a n i a d i d not c o i n c i d e , as i n many previous e x p e r i  ments'*"^, with the E f i e l d p a t t e r n . The f i r s t t e s t did. have one large t r a c k at a r a d i u s of 1.8 cm, the p o s i t i o n of maximum E f i e l d i n the c a v i t y . A p o r t i o n of the brass end w a l l , d i r e c t l y opposite the breakdown r e g i o n , was a l s o d i s c o l o u r e d . The d i s c o l o u r a  t i o n had the form of c o n c e n t r i c r i n g s of v a r i o u s c o l o u r s , mainly reds and b l u e s . The r i n g s Were c.oncentric and c i r c u l a r to a high degree. There was no d i s t o r t i o n where they i n t e r s e c t e d the c o u p l i n g h o l e . Figure 3.3 shows the breakdown on the d i s c and F i g u r e 3.4 shows the c o n c e n t r i c r i n g s . 41 3*3 Breakdown Test I I I - Glazed D i s c . I n an e f f o r t to e l i m i n a t e the surface breakdown observed i n the previous t e s t s , the side of the d i s c nearest th© coupling hoi© was sprayed, with a l e a d borate solution. Th© disc was then heated to a temperature of 430°0 for a glassing action to ooeur. Thin proe©s§ produeed a smooth gleisy surfae©? th© f i n a l glaze thick ness being between 0.001 iaeh and 0.005 ineh. During th§ high power t©it on th© dise, th© &©ig©r Counter again r§gist©r©& only background radiation. This time, hew©v©r, a blue glew=diseharg© was visibl© i n th© baek ehamb©r of th© cavity* Th© diseharg© wai f i r s t netieeabl© at about 250 Kw and iner©as©d i n brightness as th© pew©r le v e l was raised. B©tw©©n 1000 Kw and 1200 Kw th© magnetron misfired repeatedly and sparking oecurred within th© cavity. On dismantling the cavity* no major breakdown was v i s i b l e on e i t h e r th© g lazed or unglazed surface of the d i s c . There was one smal l t r a c k of reduced ceramic o r i g i n a t i n g at the edge of the d i s c , on the g lazed s i d e . The removal of the copper support ing r i n g r e v e a l e d extensive breakdown along the d i s c p e r i m e t e r . The d i s c o l o u r a t i o n occurred i n areas where the d i s c had made imperfect contact w i t h the r i n g * 42 3*4 Breakdown Test IV - Glazed Disc With Thin Aluminum Edge-Coating* The glaze on the d i s c surface was d i s s o l v e d i n n i t r i c a c i d * The reduced area was r e s t o r e d to t i t a n i a ceramic by h e a t i n g the d i s c to 850°C i n an o x i d i z i n g atmosphere, and a l l o w i n g i t to co o l g r a d u a l l y . One side of the d i s c was then r e g l a z e d . The presence of edge breakdown, even i n the t h e o r e t i c a l absence of strong E f i e l d s at the edge of the d i s c , was somewhat s u r p r i s i n g and i t was decided to i n v e s t i g a t e the use of a metal f i l m on the d i s c edge. Such a c o a t i n g provides a more uniform metal-ceramic bond than i s p o s s i b l e with only the d i s c arid metal r i n g . Many metals are s u i t a b l e f o r t h i s purpose but aluminum was chosen f o r the r e l a t i v e ease with which i t i s p o s s i b l e to evaporate, onto a s u r f a c e , a homogeneous l a y e r of uniform t h i c k n e s s * The d i s c was p l a c e d i n an evacuated j a r . I t was supported about 4 inches above a tungsten f i l a m e n t around which was wound a t h i n s t r i p of aluminum. When the pressure —4 reached 10 mm Hg or below* c u r r e n t was s u p p l i e d to the f i l a m e n t and the aluminum evaporated, a small p o r t i o n adhering to the ceramic edge* The d i s c was then rotated and a new aluminum charge was p l a c e d on the f i l a m e n t . This procedure was repeated u n t i l the e n t i r e circumference of the d i s c was coated. The f i n a l r e s i s t a n c e of the aluminum 43 f i l m * as measured across the diameter of the d i s c , was 70 ohms. This r e s i s t a n c e represents a f i l m t h i c k n e s s much l e s s than one s k i n depth of aluminum at 3000 mcs. The experimental procedure was the same as i n the previous h i g h power t e s t s * Again, the peak power was 1*2 Mw^  there was no s i g n i f i c a n t r a d i a t i o n , and a blue glow discharge was evident i n the back chamber. Subsequent i n s p e c t i o n of the d i s c r e v e a l e d a l a r g e t r e e  l i k e p a t t e r n of reduced ceramic on the back face of the d i s c * The breakdown occurred near the circumference and was l i n k e d to the edge of the d i s c by a breakdown t r a c k o r i g i n a t i n g at the metal—ceramic contact. There were a number of small p i t s i n the t i t a n i a ^surface w i t h i n the reduced area. The c a v i t y end w a l l , opposite the breakdown, was d i s c o l o u r e d by the same type of c o n c e n t r i c r i n g s t h a t were observed i n the f i r s t and second h i g h power t e s t s . There was evidence, on the d i s c edge, of some sparking between the aluminum f i l m and the copper supporting r i n g . I s o l a t e d areas of reduced t i t a n i a were present i n regions of imperfect metal-ceramic contact. This edge breakdown, however, was much l e s s severe than i n the previous t e s t . The aluminum remained completely bonded to the ceramic, even i n the r e g i o n s where sparking was most evident* In hi g h power experiments r e p o r t e d by R. Hayes"*"' 44 an edge c o a t i n g of s i l v e r was s t r i p p e d from the ceramic and deposited on the i n s i d e of the copper r i n g * F i g u r e s 3*5 and 3*6 i l l u s t r a t e the breakdown* .3*5 Breakdown Test V - Glazed Disc With Improved Aluminum Edge-Coating* The .disc was prepared as d e s c r i b e d i n s e c t i o n 3*4* For t h i s t e s t , however, a much t h i c k e r c o a t i n g of aluminum was p l a c e d on the d i s c edge. The t o t a l r e s i s t a n c e across the diameter was reduced to about 2 ohms* The a c t u a l t h i c k n e s s of the f i l m was of the order of £ the s k i n depth of aluminum at 3000 mcs. Instead of the Geiger Counter, which f a i l e d to i n d i c a t e breakdown i n previous t e s t s , an i o n guage was used to monitor the breakdown* The guage was mounted over the probe hole and was made vacuum t i g h t with an M 0 " r i n g s e a l * A sudden pressure i n c r e a s e , as a r e s u l t of i o n i z a t i o n during breakdown* r e g i s t e r s on the i o n guage and provides a means of determining the power l e v e l at which a breakdown i n the back chamber occurs* The d i s c was thoroughly decreased and p l a c e d i n the c a v i t y * The system was pumped u n t i l the pressure i n the waveguide run was 2 x lO""^ mm Hg; the back chamber —4 pressure was about 10 mm Hg* During the outgassing process each jump i n back'chamber pressure was i n d i c a t e d on the i o n guage* Once the system was outgassed, no f u r t h e r pressure change was recorded* The maximum power l e v e l f o r P i g * 3.6 Edge Breakdown - Test IV / 46 t h i s t e s t was 1*6 Mw* At.the 1 Mwlevel, power was d e l i v e r e d c o n t i n u o u s l y f o r a 30 minute p e r i o d * Dismantling the c a v i t y repealed only a. l i g h t brown s t a i n on both s i d e s of the d i s c * The d i s c o l o u r  ation was i n the form of a Tsmoke>»ring 1 about the centre of the d i s c , the average r a d i u s being between 1*8 and 1*9 cm* There were no areas of reduced t i t a n i a any^- where on the d i s c s u r f a c e * I n s p e c t i o n of the d i s c w i t h the copper r i n g removed showed only two i s o l a t e d areas where a weak discharge had occurred between the aluminum and the copper* Again, there were no sigr^.s of the aluminum p e a l i n g from the ceramic edge* 3*6 D i s c u s s i o n of R e s u l t s * The h e a t i n g of the ea.vity i n the f i r s t two t e s t s i n d i c a t e d t h at a l a r g e amount of power was d i s s i p a t e d i n the d i s c * producing the extensive r e g i o n of reduced t i t a n i a * This h e a t i n g i s a l s o a p o s s i b l e e x p l a n a t i o n of the c o l o u r e d c o n c e n t r i c r i n g s on the end p l a t e since h e a t i n g brass produces s i m i l a r e f f e c t s to those observed* I n t h i s case* however, any h e a t i n g e f f e c t on the end p l a t e would have produced a d i s t o r t e d p a t t e r n because of the presence of the c o u p l i n g h o l e * No such d i s t o r t i o n e x i s t e d * and i t i s f a r more proba,ble t h a t the coloured r i n g s were due to the; d e p o s i t i o n Of a t h i n f i l m on the 47 end p l a t e * V a r i a t i o n s i n f i l m t h i e k n e s s would produce the coloured r i n g s * The p i t t i n g a t the -ceramic surface suggests t h a t m a t e r i a l was evaporated from the d i s c d u r i n g breakdown and d e p o s i t e d on the eftd p l a t e . The breakdown probably occurred at a r e l a t i v e l y low power l e v e l as a r e s u l t of sparking at the surface; the sparking being i n i t i a t e d by contamination or some i n t r i n s i c weakness due to surface c o n d i t i o n or m a t e r i a l Composition* The sparking reduced the t i t a n i a to a lower semiconducting oxide t h a t then d i s s i p a t e d power and heated the d i s c * The h e a t i n g f u r t h e r reduced the t i t a n i a , l e a d i n g to more d i s s i p a t i o n and a g r e a t l y reduced c a v i t y Q Q* The c o u p l i n g f i e l d s c o u l d a l s o have played an important p a r t i n t h i s f i r s t breakdown. In the back chamber the pure mode should e x i s t because, i n terms of wavelength* i t i s w e l l removed from the c o u p l i n g h o l e . The same i s not true f o r the f r o n t chamber. An a p p r e c i a b l e a x i a l E f i e l d can e x i s t due to the c o u p l i n g h o l e ^ and t h i s f i e l d , i n c o n j u n c t i o n with the p o s s i b i l i t y of f i e l d emission from the sharp edges of the i r i s , c o uld I n i t i a t e a breakdown at the ceramic* The c o u p l i n g hole i's undoubtedly prone to f i e l d emission because the f i n a l diameter was reached by hand reaming; the r e s u l t being a s c a l l o p e d e f f e c t around the hole circumference* The above d i s c u s s i o n s t r o n g l y i n d i c a t e s that the 48 breakdown was not s o l e l y dependent on an E f i e l d t a n g e n t i a l to the d i s c s u r f a c e * For t h i s reason the low power l e v e l at which breakdown occurred does not y i e l d a true,, i n d i c a t i o n of the surface breakdown p o t e n t i a l of a t i t a n i a d i s c * For the t h i r d t e s t * w i t h the glaze i n h i b i t i n g the breakdown on the f r o n t face of the d i s c , i t i s reasonable to assume t h a t the f i e l d s w i t h i n the c a v i t y i n c r e a s e d u n t i l the next most v u l n e r a b l e p o r t i o n of the d i s c f a i l e d * The assumption of higher f i e l d s i s a l s o supported by the presence of the glow discharge i n the back chamber* The discharge was the r e s u l t of i o n i z a t i o n of the low pressure a i r i n the c a v i t y * Since f o r a l l t e s t s the pressure was of the same order of magnitude, the discharge i n d i c a t e s higher f i e l d strengths i n the t h i r d t e s t than i n the f i r s t two t e s t s * This i n c r e a s e d power l e v e l caused breakdown at the d i s c edge* T h e o r e t i c a l l y ^ the E f i e l d i s zero at the d i s c edge and edge breakdown should not occur* How ever* surface i r r e g u l a r i t i e s and l o s s e s i n the w a l l s w i l l produce f i n i t e f i e l d s at the d i s c perimeter. These f i e l d s can be enhanced by as much as a f a c t o r of (93 f o r t i t a n i a ) , and can cause f i e l d emission i n gaps at the metal-^ceramic contact* The r e s u l t i n g sparking can cause breakdown at the t i t a n i a edge* Trapped pockets of a i r with r e l a t i v e l y low breakdown strengths could a l s o cause 49 sp a r k i n g , i n i t i a t i n g breakdown at the ceramic. The aluminum f i l m used i n t e s t IV greatly- reduced the edge breakdown by p r o v i d i n g a uniform metal—ceramic c o n t a c t . The f i e l d s were f u r t h e r i n c r e a s e d , to the p o i n t where a major breakdown occurred on the back surface of the d i s c * The breakdown mechanisms d e s c r i b e d i n co n j u n c t i o n w i t h t e s t s I arid I I were probably also r e s p o n s i b l e f o r the surface break down i n t e s t IV, although t h i s time i t appears that the i n i t i a l h e a t i n g and sparking was a s s o c i a t e d with the edge of the d i s c • The f i n a l t e s t e s t a b l i s h e s t h a t the surface break down strength of t i t a n i a i s much higher than has been measured i n previous experiments* B, Hayes"*"^ has shown Kv that at r a d i a l E f i e l d s of between 6 and 14 — t i t a n i a cm d i s c s w i l l f a i l because of edge e f f e c t s . In t e s t V, with edge breakdown almost completely suppressed, the E f i e l d Kv st r e n g t h was in c r e a s e d to w e l l over 50 — , and no break-" • cm 7 down occur r e d . Two conclusions may be drawng ( i ) The surface breakdown s t r e n g t h of t i t a n i a i n the presence of a t a n g e n t i a l E f i e l d i s s i g n i f i c a n t l y higher than has been measured when edge e f f e c t s have i n i t i a t e d breakdown. This r e s u l t i m p l i e s t h a t , p r o p e r l y handled, t i t a n i a can be s u c c e s s f u l l y employed i n high 50 power microwave devices* ( i i ) The aluminum f i l m s u b s t a n t i a l l y improves the edge breakdown st r e n g t h of the t i t a n i a d i s c s , and i t does not peel from the surface at h i g h r * f * power l e v e l s . 4. ELECTRON MOTION IN THE S-BAND CAVITX 4*1 I n t r o d u c t i o n * The presence of s u f f i c i e n t l y e n e r g e t i c e l e c t r o n s * i n c o n j u n c t i o n with the a p p l i e d electromagnetic f i e l d * can i n i t i a t e breakdown at a d i e l e c t r i c s u r f a c e * When e l e c t r o n s impinge on the surface most of t h e i r energy i s l o s t i n e x c i t i n g e l e c t r o n s i n the d i e l e c t r i c * This e x c i t a t i o n can r e s u l t i n b r i n g i n g e l e c t r o n s from the valence band to the conduction band, some wit h s u f f i c i e n t m o b i l i t y to reach the surface and escape* The amount of t h i s secondary emission and the surface c o n d u c t i v i t y govern the magnitude of l o c a l surface charging which can produce high v o l t a g e g r a d i e n t s , p o s s i b l y r e s u l t i n g i n e l e c t r i c a l d ischarges* In a d d i t i o n to i n c r e a s e d C o n d u c t i v i t y and surface charg^ng^ s u s t a i n e d i bombardment* by an e l e c t r o n resonance or m u l t i p a c t o r e f f e c t * can cause l o c a l h e a t i n g intense enough to chemi^ c a l l y reduce the d i e l e c t r i c s u r f a c e * The present d i s c u s s i o n i s concerned with the motion of e l e c t r o n s emitted from the d i e l e c t r i c s s u r f a c e * w i t h emphasis on determining upper l i m i t s to e l e c t r o n energies* R* Haye-s^ concluded t h a t t i t a n i a c o u l d w i t h  stand bombarding energies up to 10 Kev without s t r u c t u r a l damage* hence+ such upper l i m i t s are u s e f u l i n as s e s s i n g the p a r t played by e l e c t r o n bombardment i n the breakdown* as w e l l as i n e x p l a i i i l t i g the complete absence of any d e t e c t a b l e X - r a d i a t i o n d u r i n g the h i g h power t e s t s * .4*2 Approximate Numerical S o l u t i o n by Method, of Eunge- Kutta* The c l a s s i c a l equations of motion, d e r i v e d from the f o r c e equation S - J = eE + eV x I and expressed i n c y l i n d r i c a l c o o r d i n a t e s , as given by V*H* H a y t 1 8 aret - r (6 ) 2 = | ( E r + B z « r - B^z) z = — ( E + B Q r — B »r) mv z © r where the appropriate c a v i t y f i e l d s a r e i H = -H J ( k r h i n h o c z ^ e ^ " 6 z 1 o H r = f ^ ^ C k r J c o s h a z . e , ^ ...4,2 E S = H 1 J 1 ( k r ) s i i i h a z . e ; 5 w t where y, = = JXQ' = 4it X 10~ — • 53 A s o l u t i o n was attempted by n e g l e c t i n g the H f i e l d s and l i m i t i n g the p o s s i b l e i n i t i a l values of r to 0*0184 — r Q i=z 0*0385 metres* In t h i s range J ^ ( k r ) was -»3 approximated as 11*5 x 10 . ... , , „ r r — • — — — — * An a r b i t r a r y l e v e l of EOmax = 2 9 cm w a s u ® e < ^ * a n < ^ ^ e time v a r i a t i o n was suppressed* The f o l l o w i n g equations r e s u l t e d ! r - r & 2 = 0 and •••4*3 The numerical method of RungexKutta was used and a s o l u t i o n f o r the t r a j e c t o r y of an e l e c t r o n emitted at r = 0*0184 m was obta i n e d * The time the e l e c t r o n o remained i n the f i e l d * was approximately 4 x 10""^ sec, which i s comparable to one c y c l e at 3000 me, hence, the suppression of e^a^ was not. j u s t i f i e d . Two f u r t h e r s o l u t i o n s were obtained i n c l u d i n g the time dependence* assuming t Q = 0 and t Q = T^, where T i s the p e r i o d * and r Q = 0*0184* F i g u r e 4*1 shows the r e s u l t i n g t r a j e c t o r i e s * The v e l o c i t i e s obtained* assuming no H f i e l d s , when s u b s t i t u t e d i n t o F = eY x S* showed t h a t the magnetic c o n t r i b u t i o n was not n e g l i g i b l e * In f a c t , the r e s u l t i n g v e l o c i t y i n the z»-direction was comparable to F i g * 4,1 T y p i c a l T r a j e c t o r i e s the r a d i a l v e l o c i t y , A l s o f these numerical s o l u t i o n s y i e l d e d no g e n e r a l i z e d energy i n f o r m a t i o n and another method based on the Hamiltonian of the,: motion was used 4*3 P o t e n t i a l Expressions f o r C a v i t y F i e l d s , In order to use the Hamiltonian approach to e l e t r o n dynamics, i t i s necessary to know the s c a l a r and v e c t o r p o t e n t i a l s r e p r e s e n t i n g the f i e l d q u a n t i t i e s . L e t X be the v e c t o r p o t e n t i a l and V be the s c a l a r p o t e n t i a l * thens u-H = c u r l A ««,4 ,4 E = -grad V - I 55 The c a v i t y f i e l d s are as giv e n i n s e c t i o n 4.2. Hence .i * ^ H, J, ( k r ) c o s h a z * e D f l > t - r*uH,J (kr) s i n h c t z e / ^ r k 1 1 x ' z r 1 o • i a v ? i o r 3 0. " d z i + r 1 3 r A Q r 9 r dK 1 v / - r r <70~ and i 0 * ^ H 1 J 1 ( k r ) s i n h a z * e ; ] ' a > t = i " G ( - joAg) - 1 3 7 1 9 * r 90 A s o l u t i o n of these equations gives; —B H = — ^ J ^ ( k r ) s i n h a z e ^ ^ * i g ...4.5 V = 0 where Bn = » H 0 1 4»4 The Hamiltonian and the Equations of Motion The c a v i t y f i e l d s reduce to a s i n g l e v e c t o r p o t - _ B-j^  sinttt _ e n t i a l A = — ^ J ^ ( k r ) sinhocz i ^ . <-cosfl>t The Hamiltonian of the motion* when expressed i n terms of P* the v e c t o r canonic momentum. 20 and the s c a l a r and v e c t o r p o t e n t i a l s , i s giv e n as i 1_ 2m P - eA + eV 56 where e i s the charge* S u b s t i t u t i n g f o r P and X 2 K= ST |" Pr + P 0 " e r T J 1 ( k r ) s i n h a z sinwtl -cosiftt, + P ' *•.4.6 The equations of motion are obtained from J"£ y i a the canonic equations of Hamilton! * i = a*. and. ax .•.4.7 where ^1 = r * ^2 = ®* q 3 = z * Hencei (1) (2) '© r = _r m !» eB, J , ( k r ) * 1 s i n t o t km sinhaz -coswt (3) •7 m (4) P r dr eB sinwt' -^coswt ...4.8 (5) PQ= 0 and P^ = constant 57 (6) P 2m eB, -g— J ^ ( k r ) s i n h a z simtt-tA -cosftxt M a n i p u l a t i n g equations 4*8 and making a time.-' average approximation, as v i l l be done i n subsequent s e c t i o n s , y i e l d ? % z V km/ 2 J-, 2(kr) 2 ~ 1 ' . ' i - — • s i n h az 4 ...4.9 Thus* z i s an a c c e l e r a t i o n away from the surface of the d i s c , towards regions of, weaker E f i e l d , and the f i n a l e l e c t r o n energies w i l l be smaller than i f the motion had been confi n e d to the plane of the d i s c . Upper l i m i t s to t r a n s v e r s e v e l o c i t i e s * t h e r e f o r e , can be obtained by e l i m i n a t i n g the z-dependence of A, thereby c o n f i n i n g the e l e c t r o n s to the t r a n s v e r s e plane. Sections 4.5 and.4*6 u t i l i z e t h i s s i m p l i f i c a t i o n . 4*5 E l e c t r o n Emitted at I n s t a n t of Maximum E^-Time- Ayerage Approximation* The z-independent v e c t o r p o t e n t i a l becomes? B n and A Q = J ^ k r j s i n w t P Q e B l ^ J.Ckr) © = ...4.10 sincot ...4.11 mr km 58 P r dr k^ 4-^ J i ( k r ) s i n a , t j I 2 The problem i s f u r t h e r s p e c i f i e d by the i n i t i a l c o n d i t i o n s * Assume © = o« r = r at t = t = o. • r • o o Equations 4*11 show that f o r t h i s case, Fg = o, and. P r = " ^ [ ( T 1 ) J 1 ( k r ) s i n t t t 2. 1_ 2m hence 8 r , eB-2 , , ...4.12 At t h i s p o i n t the assumption i s made that tot V a r i e s r a p i d l y enough to j u s t i f y the use of the time- 2 1 averaged a c c e l e r a t i o n , and s i n tot i s r e p l a c e d by » Therefore r 8r eB » » » r r 1 d • 2 2 dt 1" a)nd r 2 + 2 ^ ( r ) = K Q* but r = 0 at r = .r eB eB, ...4.13 59 Hence and ...4.14 The e v a l u a t i o n of t h i s i n t e g r a l i s s i m p l i f i e d by l i n e a r i z i n g J ^ k r ) over the range of i n t e r e s t , 0.0184 — r - 0,0385 metres*' A s u f f i c i e n t l y accurate approximation i s J 2 ( k r ) ~ 0*65 - 16.9r ...4.15 0 1 2 3 4 r (cms) 2 P i g * 4 » 2 J i (kr) Approximation 60 Fig u r e 4*2 i l l u s t r a t e s the nature of t h i s approximation. S u b s t i t u t i n g equation 4*15 i n 4.14, <5(5fc) \ — — TC = tflSr) ^ J [ J l i k r o ' " °*65 + l 6* 9 rJ 1 dr (a + b r ) 2 e B 1 ) 16.9 J 2 ( k r ' ) - 0.65 + 16.9r 1 o' Therefore f o r r = 0.0184 metres ...4.16 Now squaring both s i d e s of equation 4.16 y i e l d s : 2 mk eB, , r ( t ) = 2*11 (—~) \* + 0.0184 *.*4.17 R e f e r r i n g back to equation 4.13 f o r r f i t i s p o s s i b l e t o e s t a b l i s h an upper l i m i t to the r a d i a l v e l o c i t y * 2 2 eB, , eB n ^ Therefore at the c a v i t y w a l l * r = r = 0,0385 m* 61 eB 2 ( r 2 ) = \ ( — f ) J 2 ( k r ) as J 2 ( k r ) = 0 v 'max 2 • mk ' l v o' 1 Hence eB The r a d i a l v e l o c i t y . w i l l be maximum when both J , ( k r ) and B, are maximum, B, i s a maximum when 1 v o' 1 1 the t o t a l a v a i l a b l e power of 2Mv i s f e d to the c a v i t y , o perating w i t h Q Q = 2000, and J ^ ( k r Q ) i s a maximum 4 i . = 5*1 x 10 x u, , lmax r o when r = 0.0184 m. B, = 5*1 x 10 x u, • Then |r|£ 48 x 10 6 m/s The t a n g e n t i a l v e l o c i t y component, r©, i s given ass eB, and For r3 = (-^-)* ^ ( k r ^ s i n t o t ...4.19 r = 0.0184 m* B 7 = u x 5.1 x 10 4 o T 1 ro |r©| < 67 x 1 0 6 m/s In the a c t u a l resonant c a v i t y there w i l l a lso be an a x i a l component of v e l o c i t y * z* since the f i e l d s are 62 dependent on z» Equation 4»9 enables an estimate to' be made of z* z 3z eB 2 ^ 2 ( k r ) 1 ;| ...4.' ( — ) K km' s i n h az The s o l u t i o n of t h i s equation i s s i m p l i f i e d by l i n e a r i z i n g 2 •» sinh az i n such a f a s h i o n t h a t z i s always greater than the a c t u a l z* This may be adcomplished by t a k i n g 2 s i n h az ^ 107z f o r 0^ a z ^ 1, Therefore S = ^ J x 2 ( k r ) , J x 2 ( k r ) = 0.65 - 16 . 9 eB eB, z ^-26.8 (-ri) km' 0*65 - 16.9r Prom equation 4*17, z ~ -26.8 ( ^ ) knr eB, < . 0,34 - 35.7 (-^) t 2N eB, 2r eB, 0,34t - 11.9 (-^) t + C ...4.20 Hence but z = 0, t = t = 0 , t h e r e f o r e C = 0 o ' eB, eB, 2 1 0»17t 2 - 2.98 (-^i) t + .0129' ...4.21 63 I n s p e c t i o n of equation 4*21 i n d i c a t e s t h at the t r a n s i t time of the e l e c t r o n from the d i s c to the c a v i t y end w a l l i s . t^< 5 x 10"° 1 0 sec* and r e f e r r i n g to equation 4*20$ 2 eB |z | < 45. 5 ( r * ^ ) x l O " " 1 0 ...4.22 |z| < 58 x 10 6 m/s The t e r m i n a l v e l o c i t y * U^, of the e l e c t r o n s can be obtained by the v e c t o r sum of the component v e l o c i t i e s . Thus |UT| < ( 5 8 2 + 4 8 2 ) 2 x 10 6 IUTI< 75.2 x 10 6 m/s Equating energies eV = | m |U T| 2 and v = 75.2 x 75.2 x 1 0 1 2 „ 1 6 0 0 0 v o l t s , 5.93 x 5.93 x 1 0 1 0 ' Hence* the t e r m i n a l energy of an e l e c t r o n emitted from the d i s c at an i n s t a n t when the E_ f i e l d i s a 64 maximum i s l e s s than 16,0 Key, 4 #6 E l e c t r o n Emitted at I n s t a n t of Minimum E^-Time- Average Approximation. T Consider the e l e c t r o n r e l e a s e d at t + -r , which o 4 1 i m p l i e s th a t * Therefore -B 1 A Q = —^r~ J-j^ (kr)costtt P Q eB, J , ( k r ) 9 = -^7r + (-pi-) — — c o s t t t 2 v km' r mr P =~2- r 3r 1_ 2m © eBj^ —^— (kr)costot ...4.23 and r where © m 2 r 3 »r eB, 2 J , 2 ( k r ) „ P Q eB, J, (kr) (-^) ^ - 5 — c o s V t + -£ . - V - c o s a t eB -(-T^-)r J, (kr ), from © = at t = t = 0. v k ' o l o ' y o Again take the time-average of r * Note, however, t h a t i n t h i s case the approximation i s not as good as before because of the coswt term appearing i n V. 65 Thus 2 2 r = m 2 ^ ' ^ km ...4.24 and m r o r i* + ^ • 2 ft,) - K ( m r * 2 S u b s t i t u t i n g f o r r = r Q , r = O at t Q = 0, K = p 2 eB 2 eB 2 O 2^. , _ X U 4 . 1__ : m r km Km o and 9 2 2 2 - eB, z 0 , e B * ~ eB, i r * 0 r 2 = J 2 ( k r ) - J 1 2 ( k r ) - (-^) J 2 ( k r Q ) 2 km 1 0 ^ km 1 km r" 1 1 0 ...4.25 dr  T T 2 2 P 2 o eB, 9 i e B i o eB, r/ 0 r = ^ i t ...4.26 Using equation 4.15 and ~ = 0.57 x 10 4 -.14.8 x 1 0 4 r r ^ f o r 0.0184 ^ r ^ 0.0385 metres, equation 4.26 becomes: 2 66 eB, 2 0 r = 6.36 t + 0.0184 ...4.27 As i n S e c t i o n 4.5 r i s a maximum when r =0.0184 mt o B, = u x 5.10 x 10 4 webers and r = 0.0385 m, the 1 ro — - ~ 2 — max m c a v i t y w a l l . Hence a R 2 p. 2 max 2 X mk' l v o 2 2 m r max Therefore - eB 2 ' eB 2 ^ r max ...4.28 and [ r | ^ 75 x 10 6 m/s The t a n g e n t i a l v e l o c i t y can be found from equations 4*23 as: -eB, r eB, r G = mk'* r 2 " * J l < k r o > + ( ^ ) J 1 ( k r ) c o s W t hence K a x K 2 (-mf> Jl< k ro> — 4 - 2 9 |r0|< 132 x 1 0 6 m/s 67 The a x i a l v e l o c i t y can be approximated i n the same f a s h i o n as i n s e c t i o n 4*5* us i n g equation 4.27. eB, ' eB, A p 0*34 - 1 0 7 . 5 ( ^ ) t * eB, ' .J * .26,8(-jl> eB - 0*34t - .36 ( - ^ ) V eB, ' , a -26.8 (^ i ) and t 2 5 x 1 0 ~ 1 0 sec* r 0 . 1 7 t 2 - 9 ( - i ) t 4 + 0.0129 eB |z| < 3 5 ( ^ 0 x 1 0 " 1 0 ...4.30 |z| < 44*5 x 10 b m/s Hence, the te r m i n a l v e l o c i t y i s given as: l r | - 75 x 10 6 m/s |z|^ 44*5 x 1 0 6 m/s |r©T| < 31*5 x 10° m/s 2x i |UT| < ( 7 5 * + 44.5* + 31.5*) 2 x 10° m/ |UTI < 93 x 10 b m/s 68 Hence '93 x 10 6  2 5.93 x 1 0 5 24,000 v o l t s The t e r m i n a l energy of an e l e c t r o n emitted from the d i s c at an i n s t a n t when the E Q f i e l d i s a minimum i s l e s s than 25 Kev. 4*7 E s t i m a t i o n of E r r o r Introduced by The Assumption of Time-Averaged Values* 1 Another approach to the s o l u t i o n of equations 2 4*8 i s p o s s i b l e , t h a t i s to l i n e a r i z e ( k r ) , J ^ ( k r ) , —^ before i n t e g r a t i o n * The r e s u l t i s a l i n e a r r d i f f e r e n t i a l equation with constant c o e f f i c i e n t s with the time dependence i n t a c t * When reasonable l i n e a r i z a t i o n s are taken, an estimate of the maximum err/or a s s o c i a t e d w i t h the time- average approximation is. / p o s s i b l e . Since the l a r g e s t e r r o r w i l l be a s s o c i a t e d with the, motion d i s c u s s e d i n S e c t i o n 4.6, t h i s case w i l l be d e a l t w i t h . R e c a l l i n g equations 4.23, P 2 2 3" m r *3r eB, 2 J , (kr) 0 P Q eB, J , ( k r ) (—-) — — — c o s ^ t t t + -2*-_A._± coswt km 2. m km r . The f o l l o w i n g l i n e a r i z a t i o n s are used: - 3 = 1.75 x 1 0 5 - 4.17 x 1 0 6 r r 69 J-^Ckr) = 0*65 - 16.9r 0% (kr) — = 60 - 15S0r r where 0.01841 r ^ 0.0385m. Therefore p 2 p 2 e B 2 - 9 x 1.75 x l 0 5 - 4 : x 4*17 x 1 0 6 r + i ^ ( _ i ) c o s 2 « t r m m 2 km P a eB + — ( - r ^ ) 1580 coswt mx km' Hence r = 0*051 - 0.0326cosw t - 0.575 x 10~ 4cos2ttt r —4 - 4.66 x 10 coswt r = 81 x 10 6sinfi> rt + 2.5 x 10 6sin2wt + 10 x 10 6sincot Time-averaging corresponds to dropping terms i n (<ot) from r and r . The r e s u l t i n g e r r o r i s l e s s than 3 percent i n r and l e s s than 16 percent i n r . Thus, to a reasonable degree of accuracy, time—average f o r c e s can be used. 4*8 Lorentz Time-Average Force* A charged p a r t i c l e of e i t h e r s i g n i n a non-uni-70 form f i e l d experiences an a c c e l e r a t i o n towards the p o s i t i o n of sma l l e s t e l e c t r i c f i e l d s t r e n g t h . The a c c e l e r a t i o n i s due to the time—averaged Lorentz f o r c e J t , 21.22 * and i s given by * s -e Arm' grad I E I Sl2 ...4.31 where e i s the charge, m i s the mass and E i s the e l e c t r i c f i e l d v e c t o r . The E f i e l d i n the c a v i t y i s e n t i r e l y t r a n s v e r s e and given by» E e = ^ - ^ - ^ ^ ( k r j s i n h a z . e 0 " " 6 Hence « 2 B 2 E j 2 = 1 - J 1 2 ( k r ) s i n h 2 a z ...4.32 Prom 4.31 f = -eV 4mk 2 I 1 r J—- J ^ 2 ( k r ) s i n h 2 a z + XOr5© J-^ 2(kr) s i n h 2 a z ] + i" I" z dz l 2 ( k r ) s i n h 2 a z ] -e\2 4mk 2 r d r 2 ( k r ) s i n h ^ a z i _ z | ^ { j 1 2 ( k r ) s i n h 2 a z j See Appendix 1. 71 2 B 2 -e B 2mk — sinh 2oczJ^ (kr) J (kr) r — J n ( k r ) 0 k r x 2„ 2 -e B 2mk 1 2 — —s- ocsinhctz coshaz J , ( k r ) i •••4,33 Thus* charged p a r t i c l e s i n the c a v i t y experience a net for c e a c c e l e r a t i n g them away from the surface of the d i e l e c t r i c d i s c towards the c a v i t y end w a l l . I f the p a r t i c l e s are r e l e a s e d at TQ > 0,0184m, they w i l l d r i f t r a d i a l l y outwards and i f r Q ^  0,0184m, they w i l l be a c c e l e r a t e d towards the a x i s * F i g u r e 4.3 i l l u s t r a t e s the type of motion. i F i g . 4.3 Time-Averaged E l e c t r o n Motion 4*9 M u l t i p a c t o r . Secondary e l e c t r o n resonance e f f e c t s , or m u l t i - p a c t o r s * are of two b a s i c types8 two-surface, and singles-surface* as r e c e n t l y proposed by P r i e s t and T a l c o t t 2 ^ . The two—surface m u l t i p a c t o r occurs when e l e c t r o n s from one surface are a c c e l e r a t e d towards the second and a r r i v e there at such a time t h a t any secondaries emitted w i l l be* i n t u r n * a c c e l e r a t e d towards the f i r s t s u r f a c e * I f the a r r i v a l energy of the e l e c t r o n s i s w i t h i n the range f o r which the secondary emission c o e f f i c i e n t {6) i s g r e a t e r than u n i t y , the number of o s c i l l a t i n g e l e c t r o n s w i l l i n c r e a s e r a p i d l y * The e f f e c t i s c r i t i c a l l y dependent on the f i e l d , the gap len g t h and the phase angle of the f i e l d at which an e l e c t r o n i s emitted. In the present c a v i t y i t has been e s t a b l i s h e d t h a t e l e c t r o n s w i l l be a c c e l e r a t e d away from the surface of the d i s c , and hence, a two—surface m u l t i p a c t o r between the d i s c and end w a l l i s v i r t u a l l y i m p o s s i b l e * S i n g l e — s u r f a c e m u t l i p a c t o r i s dependent on f r e e e l e c t r o n s *dancing' on the s u r f a c e * The e l e c t r o n s t r a n s  f e r energy from the f i e l d to the s u r f a c e , causing more e l e c t r o n s to be r e l e a s e d by secondary emission. Secondarie are i n i t i a l l y c a r r i e d away from the sur f a c e by t h e i r emission v e l o c i t i e s but are retu r n e d by a r e s t o r i n g f o r c e due to surface charging o r the presence of a t a n g e n t i a l magnetic f i e l d * The t e s t c a v i t y has E and. H f i e l d 73 components t a n g e n t i a l to the s u r f a c e * I f an e l e c t r o n i s emitted with a f i n i t e a x i a l ' v e l o c i t y at a time such t h a t the H • f i e l d produces a p o s i t i v e r e s t o r i n g f o r c e a m u l t i  p a c t o r could occur* p r o v i d e d t h a t the t o t a l time the e l e c t r o n remains i n the f i e l d i s much l e s s than one r * f • c y c l e * This time i s given by * In other words 22L-<r<3*3 x 1 0 t 1 ° sec, ..*4.34 eo-j^  The i n e q u a l i t y 4*34 would imply that B = 10B , where B i s the maximum B, f i e l d p o s s i b l e i n the c a v i t y , and max 1 * J r thus t h i s type of m u l t i p a c t o r i s r u l e d out. I f , on the other hand* the r e s t o r i n g f o r c e i s due to a p o s i t i v e surface' charge i t i s p o s s i b l e f o r the t e s t c a v i t y to s u s t a i n a s i n g l e surface m u l t i p a c t o r on the ceramic d i s c . This e f f e c t i s p o s s i b l e over a wide range of f i e l d s trengths which w i l l enable the bombarding e l e c t r o n s to give <S > 1» 4*10 Summary. The f o r e g o i n g a n a l y s i s i n d i c a t e s t h a t the f o l l o w i n g hypotheses and con c l u s i o n s are v a l i d s l ) The maximum energy of an e l e c t r o n s t r i k i n g any metal surface i n the c a v i t y i s l e s s than 25 Kev* Since the o p e r a t i n g Q i s l e s s than the Q o used i n the c a l c u l a t i o n s , and because the maximum power l e v e l f o r the hig h power t e s t s i s always l e s s than the 2Mw peak* i t i s probable t h a t the e l e c t r o n energies are not i n excess of 10 Kev* 74 Bombarding energies of t h i s order are not s u f f i c i e n t to give s i g n i f i c a n t x - r a d i a t i o n e x t e r n a l to the c a v i t y . E l e c t r o n s experience a d r i f t f o r c e away from the d i s c surface and hence i t i s not p o s s i b l e f o r them to r e t u r n to the surface with h i g h e n e r g i e s . As a r e s u l t d i r e c t bombardment by en e r g e t i c e l e c t r o n s does not have an important p a r t i n the observed breakdown. There can be no two—surface m u l t i p a c t o r , again because of the a x i a l d r i f t away from the d i s c s u r f a c e , A s i n g l e - s u r f a C e m u l t i p a c t o r i s p o s s i b l e with a p o s i t i v e surface charge. The resonance, however* i s not s e l f — s t a r t i n g , I t r e q u i r e s the/presence of a bombarding strekm of e l e c t r o n s having energies t h a t give 1> so as to b u i l d up the necessary p o s i t i v e charge. Such a stream i s u n l i k e l y i n the t e s t c a v i t y , and hence, while p o s s i b l e * t h i s m u l t i p a c t o r should not p l a y a major r o l e i n the breakdown. 75 5 , CONCLUSIONS Ceramic breakdown at microwave f r e q u e n c i e s , under the i n f l u e n c e of p u r e l y t a n g e n t i a l E f i e l d s , was i n v e s t i g a t e d u s i n g an S-Band c a v i t y o p e r a t i n g i n an mode* The d i s c u s s i o n of e l e c t r o n motion i n the c a v i t y showed t h a t e l e c t r o n energies were of the order of 10 Kev* and t h a t the f i e l d s caused charged p a r t i c l e s to d r i f t away from the d i s c s u r f a c e * Hence, e l e c t r o n bombard ment and m u l t i p a c t o r are not s i g n i f i c a n t l y important as breakdown mechanisms i n t h i s c a v i t y , a f a c t that i s supported by the absence of d e t e c t a b l e X - r a d i a t i o n d u r i n g the h i g h power t e s t s . The s e r i e s of h i g h power t e s t s showed t h a t , by suppressing edge e f f e c t s * p a r t i c l e bombardment, and m u l t i p a c t o r , the surface breakdown s t r e n g t h of t i t a n i a i s g r e a t l y i n c r e a s e d . The f i n a l t e s t , u s i n g a l l a v a i l a b l e power, f a i l e d to cause breakdown of the t i t a n i a . The Kv E f i e l d , i n t h i s case, was i n excess of 50 — , a marked improvement over the 6 to 14 ~ measured by R» H a y e s 1 0 * This s i g n i f i c a n t i n c r e a s e i n the power ha n d l i n g c a p a b i l i t i e s of the t i t a n i a * i n the H q 1 c a v i t y , suggests the use of the H >^ mode i n other h i g h power microwave a p p l i c a t i o n s * For example* h i g h power microwave windows t h a t commonly use the H -. mode co u l d handle much higher 76 powers i f they _yere designed to operate i n the fi^ mode. The f i r s t aluminum f i l m s u b s t a n t i a l l y reduced the edge breakdown and adhered w e l l to the t i t a n i a . I n c r e a s i n g the f i l m t h i c k n e s s provided a f u r t h e r improvement. I t i s p o s s i b l e t h a t * i f the depth of the f i l m i s made grea t e r than a s k i n depth, the edge problem can be e l i m i n a t e d * thus making complicated metal—ceramic brazes unnecessary. The work Undertaken i n t h i s t h e s i s p o i n t s out se v e r a l t o p i c s f o r f u r t h e r i n v e s t i g a t i o n . These ares ( i ) the study of a h i g h power microwave window ope r a t i n g i n an mode, ( i i ) the t e s t i n g of the aluminum bond under the i n f l u e n c e s of r a d i a l and a x i a l E f i e l d s , and ( i i i ) t h e . i n v e s t i g a t i o n of t h i n m e t a l l i c f i l m s on the surface of the ceramic* 77 APPENDIX 1: LORENTZ—TIME-AVERAGED FORCE The Lorentz f o r c e on an e l e c t r o n i s f = e(E - Y x B ) e e ...(1) where .E •• B G are the time v a r y i n g f i e l d s at the instantaneous p o s i t i o n of the e l e c t r o n s * Assume! ( i ) t h a t the e l e c t r o n motion i s such t h a t over one r * f * c y c l e the e l e c t r o n sees a n e a r l y uniform f i e l d * ( i i ) t h a t the motion w i l l be o s c i l l a t o r y with angular frequency approximately it>f the angular frequency o f the f i e l d s * The f i e l d s can be r e w r i t t e n i n terms of a displacement, Ar$ from the e q u i l i b r i u m p o s i t i o n where the f i e l d s are r e s p e c t i v e l y E q and B q » Equ a t i o n ( l ) becomes? E q +_ ( A r « y ) E O + eVx B Q + ( A r - ^ ) B Q * * • (2 ) I f to a f i r s t order the term V x B can be e ne g l e c t e d * a = — E and E ~ E where a i s the a c c e l e r a t i o n ° * m e e o and E Q = E e 3 ' ^ . Then i 2 A = d t 2 m and Ar = dt m»j ~ e E 2 o ...(3) S u b s t i t u t i n g t h i s r e s u l t i n equation (2) y i e l d s * f 1 = - 2 rm e /T=> ( E o ^ E 0 + e E Q .,••(4) Therefore f, = -e 1 " \m2 + eE But Hence -e 1 ~ ™ A 2 as + eE ...(5) o £V< W + E o x St C o n s i d e r i n g f 2 g i v e s : f 2 = e(V x B Q) + eV x ( A r ' ^ B ^ but Y = ^ at j. and the second term i s small compared to the f i r s t * hence* / e 2 ^ E o ? 2 = m ? ^ X 5 ° ...(6) Therefore * ^e f = mw + eE ...(7) Taking the time-average of t h i s e x p r e s s i o n y i e l d s 8 2 T _ 1. e' 4 ^ 2 mw VlEl2 ...(8) This f o r c e i s known as the Lorentz-time-averaged f o r c e . 80 REFERENCES 1, Jasberg, J * and Lebacqz, J»V,, Advances i n Vacuum Science and Technology, v o l , 2, p, 667, 1958. 2, Lebacqz, J.V., Jasberg* J , , Shaw, H.T., and Sonkin, S., Proc. I.E.E. p a r t B, v o l , 105, s u p p l . 11, p. 617, 1958. 3, Sperry Gyroscope Company (New T o r k ) , Report No. NA-8240-8182-1 to 4, 1960, " I n v e s t i g a t i o n of M i c r o - "wave Window F a i l u r e Mechanisms and t h e i r E l i m i n a t i o n " , 4, E l e c t r i c a l and M u s i c a l I n d u s t r i e s L t d . (U.K.) Report No. HP/271* I960. 5, P r e i s t , D.H. and T a l c o t t , B.C., American Ceramic Soc. B u l l . , v o l . 38, p. 99, 1959. 6, Walker, G.B. and Lewis, E,L., Nature, v o l . 181, p. 38, 1958. 7, Walker, G.B. and L u t h r a , S.P,, "Use of Ghost Modes to Determine D i e l e c t r i c Constant and Loss Tangent", 1.E.E. Conference on Microwave Measurement Tech niques. Sept. 1961, 8, Walker, G.B., Report M.L. 2. E l e c t r i c a l E n g i n e e r i n g , U.B.C., Nov. 1960. 9, Lut h r a , S.P*, Ph.D. T h e s i s , U n i v e r s i t y of London. 10. Hayes, R., "Phenomena A s s o c i a t e d w i t h E l e c t r i c a l Breakdown at C e r t a i n D i e l e c t r i c Surfaces i n Vacua", Report No. M.L. 4ECRDC Pro.iect T54. Sept. 1961, 11. Lamont, H.R.L., Waveguides. Methuen's Monograms, John Wiley and Sons, Inc., 1959, 12. Shnurer, F., "Design of Aperature-Coupled F i l t e r s " , PGMTT, p. 238, Oct. 1957. 13. Beihe, H.A., "Lumped Constants f o r Small I r i s e s " , MIT 'RAD LAB R. No. 43-22. pp. 34-39. 14. Ramo, S. and Whinnery, H.R.f F i e l d s and Waves i n Modern Radio. John Wiley arid Sons, Inc., 1956. 15* Barlow, H.A. and C u l l e n , A.L., Microwave Measurements, p. 81, Constable and Company L t d . , 1950. 81 16. Gintzon, E.L., Microwave Measurements, McGraw H i l l Company Inc., 1957. 17. Reich, H.J., Ordung, P.F,, Krauss, H . L . , and Ska l n i k , J.G., Microvave Theory and Techniques, p. 452, Van Nostrand, 1953. 18. Hayt, W.H., En g i n e e r i n g E l e c t r o m a g n e t i c s , p. 295, McGraw H i l l Co. Inc., 1958. 19. Hildebrand, P.B., Advanced C a l c u l u s For ApT>lications, p. 102, P r e n t i c e - H a l l , Inc., 1962. 20. G o l d s t e i n , H., C l a s s i c a l Mechanics. Addison-Vesley P u b l i s h i n g Company Inc., 1959. 21. Boot, H.A.H., S e l f , S.A., and R.-Shersbey-Harvie, R.B., J . E l e c t r o n i c s and C o n t r o l , v o l . 4, p. 424, 1958. 22. Gapanov, A.V., and M i l l e r , M.A., J . Exp. Theoret. Phys., v o l . 7, p. 168, 1958. 23. P r i e s t , D.H. and T a l c o t t , R.C., I.R.E. Tra n s a c t i o n s E.D.. v o l . 8, p. 243, J u l y 1961. 

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