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UBC Theses and Dissertations

Electron paramagnetic resonance of heavily-doped n-type silicon Quirt, John David 1972

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THE DESIGN OF THE RF SYSTEM FOR THE  TRIUMF CYCLOTRON  by'  ANTONIN PROCHAZKA D i p l . Ing., T e c h n i c a l U n i v e r s i t y o f Prague, C z e c h o s l o v a k i a , 1967  A t h e s i s submitted 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 the degree o f  Doctor o f P h i l o s o p h y  i n the Department of Physics  We accept t h i s t h e s i s as conforming required  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1972  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 of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that copying or p u b l i c a t i o n  of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department of  P h  y  s i c s  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  D  a  t  e  March 1, 1972  ABSTRACT The b a s i c d e s i g n of the r e s o n a t o r system, c o u p l i n g loop assembly and resonant  t r a n s m i s s i o n l i n e f o r the TRIUMF i s o c h r o n o u s  d e t a i l and model.  the  cyclotron i s studied i n  the beam-resonator RF f i e l d i n t e r a c t i o n i s e s t i m a t e d u s i n g a  simple  The matching t e c h n i q u e s which w i l l a l l o w a maximum power t r a n s f e r  from the power tube stage to the Dees are d i s c u s s e d .  Two  major computer  programs have been w r i t t e n and are used to determine the parameters of the system n e c e s s a r y  i n o r d e r to o b t a i n a d e s i r e d impedance match and  v a r i o u s resonant  f r e q u e n c i e s of the RF  The to v e r i f y  to f i n d  RF the  system.  r e s u l t s of a l a r g e number of e x p e r i m e n t a l  t e s t s c a r r i e d out i n o r d e r  the r e s o n a t o r e l e c t r i c a l c h a r a c t e r i s t i c s , to i n v e s t i g a t e the  r e s o n a t o r frequency  t u n i n g and  the r e s o n a t o r mechanical  c o n s t r u c t i o n around  the c e n t r e p o s t , and to apply the proposed matching technique  to the TRIUMF  RF system, are g i v e n . The  r e s u l t s from the p r o t o t y p e r e s o n a t o r segments and  l i n e t e s t e d at h i g h power b o t h  the t r a n s m i s s i o n  i n a i r and under h i g h vacuum i n d i c a t e t h a t the  RF system performance i s compatible  with  ii  the c y c l o t r o n o p e r a t i o n .  C O N T E N T S Page I.  II.  INTRODUCTION  1  1. 2. 3.  1 3 5  TRIUMF C y c l o t r o n i n G e n e r a l TRIUMF C y c l o t r o n RF System C r i t i c a l RF Parameters  THEORETICAL CALCULATIONS 1.  Resonator B a s i c Parameters 1.1 1.2 1.3 1.4 1.5 1.6  2.  2.2 2.3 2.4 2.5  4.  5.  6.  7.  and R e l a t i o n s  C h a r a c t e r i s t i c Impedance T i p Loading Capacity Resonant Frequency Resonator Power Loss Energy S t o r e d i n Resonator Resonator Q u a l i t y F a c t o r  RF T o l e r a n c e s 2.1  3.  9  Frequency Detuning due t o E v a c u a t i o n of the Vacuum Chamber Detuning of Resonator due to Temperature V a r i a t i o n V o l t a g e V a r i a t i o n due to Temperature V a r i a t i o n Frequency Detuning due t o E l e c t r o s t a t i c and Magnetic F o r c e s T o l e r a n c e s to be Met  9 9 10 11 12 14 15 18 18 19 21 22 25  V o l t a g e Breakdown  27  3.1 3.2 3.3  27 29 32  Sparking M u l t i p a c t o r i n g Ranges Rate of R i s e of Resonator V o l t a g e  Frequency Tuning  40  4.1 4.2 4.3 4.4 4.5  40 40 41 42 44  Root S h o r t i n g Plane Motion C a p a c i t i v e E f f e c t Near the A c c e l e r a t i n g Gap Ground Arm T i p D e f l e c t i o n Tuning Bellows at the Root T h i r d Harmonic Tuning Diaphragms  Resonator M o d i f i c a t i o n s  49  5.1  Extreme End Segments  49  5.2  C e n t r a l Region Segments  50  Beam L o a d i n g  54  6.1 6.2  Beam-RF F i e l d I n t e r a c t i o n Beam Induced V o l t a g e  54 61  6.3  C o n c l u s i v e Remarks  64  C o u p l i n g Loop Assembly  68  7.1 7.2 7.3  68 69 70  Power Loss i n the Vacuum S e a l Power L o s s i n A d j a c e n t Areas Power Loss i n the C o u p l i n g Loop iii  Page 8.  Transmission Line  72  9.  Lumped Parameter R e p r e s e n t a t i o n  75  9.1 9.2 9.3 9.4  75 76 80  10.  III.  Resonator Lumped Constants R e p r e s e n t a t i o n of the C o u p l i n g Lumped Constant R e p r e s e n t a t i o n of a Resonant L i n e R e p r e s e n t a t i o n o f the Whole System i n Terms of Lumped Parameters  Resonant O p e r a t i o n of the RF System  83  10.1 10.2 10.3  83 88 91  O p e r a t i o n a t the Maximum Power T r a n s f e r Other Resonances of the System P o s s i b l e Operating Conditions  EXPERIMENTAL TESTS  95  1.  Measurement of Resonator  2.  Frequency Tuning 2.1 2.2 2.3 2.4 2.5 2.6  3.  Parameters  97 99  Tuning Stub Ground Arm T i p D e f l e c t i o n C y l i n d r i c a l Capacitors Capacitive Plates I n d u c t i v e Loops at the Root T h i r d Harmonic Tuning Diaphragms  V o l t a g e and Frequency V a r i a t i o n s due  Resonator 4.1 4.2  5. IV.  107  Modifications  108  Extreme End Segments C e n t r a l Region Segments  Resonant L i n e as a Matching  Network  CENTRAL REGION CYCLOTRON 1.  Low  108 108 111 117  Power L e v e l  117  1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8  2.  99 100 101 102 102 103  to M e c h a n i c a l  Misalignments 4.  82  Quality Factors Resonator F r e q u e n c i e s F i n e Frequency Tuning Coarse Frequency Tuning RF Probes C o u p l i n g Loop T r a n s m i s s i o n L i n e and Resonances of the System Comparison of R e s u l t s from Lumped and D i s t r i b u t e d Parameter R e p r e s e n t a t i o n s High Power L e v e l  127 129  2.1 2.2 2.3 2.4 2.5  129 131 131 133 134  Sparking and M u l t i p a c t o r i n g RF Contacts V o l t a g e and Frequency S t a b i l i t y C o u p l i n g Loop and T r a n s m i s s i o n L i n e RF Power A m p l i f i e r  iv  117 118 119 120 122 123 124  Page V.  SUMMARY AND CONCLUSIONS  135  References  137  Figures  138  Appendix A.  Appendix B.  Appendix C.  Centre Region C y c l o t r o n RF System Parameters the RF Fundamental  209  Centre Region C y c l o t r o n RF System Parameters the RF T h i r d Harmonic  222  Main C y c l o t r o n RF System Parameters - the RF Fundamental  229  v  LIST OF TABLES Page I. II. III.  IV.  T o l e r a n c e s on the RF parameters  8  Beam c h a r a c t e r i s t i c s  8  Percentage r e s o n a t o r frequency change vs temperature changes o f r e s o n a t o r p a n e l s  20  Resonator v o l t a g e v a r i a t i o n caused by temperature variation  22  V.  P e r m i s s i b l e l e f t - r i g h t v o l t a g e asymmetry  26  VI.  P e r m i s s i b l e top-bottom v o l t a g e asymmetry  27  VII.  Kilpatrick's  criterion  a p p l i e d t o TRIUMF r e s o n a t o r  geometry  29  VIII.  T h r e s h o l d m u l t i p a c t o r i n g v o l t a g e s f o r the RF fundamental  31  IX.  T h r e s h o l d m u l t i p a c t o r i n g v o l t a g e s f o r the t h i r d harmonic  31  Percentage frequency change vs p o s i t i o n of t u n i n g b e l l o w s  43  C a l c u l a t e d gaps and power l o s s f o r a m o d i f i e d extreme end segment Resonator power l o s s f o r a g i v e n RF v o l t a g e amplitude  50 60  X. XI. XII. XIII.  XIV.  XV.  XVI.  Computed and measured frequency s h i f t s caused by t u r n i n g the diaphragms from the h o r i z o n t a l t o the v e r t i c a l position  106  C a l c u l a t e d and measured dimensions o f the f u l l - s c a l e s i n g l e resonant s e c t i o n and the r e s o n a n t l i n e  112  C a l c u l a t e d and measured e l e c t r i c a l c h a r a c t e r i s t i c s o f the f u l l - s c a l e s i n g l e resonant s e c t i o n and the resonant line  113  Q u a l i t y f a c t o r and power absorbed vs tube s i m u l a t i n g resistance  114  XVII.  Measured  and computed q u a l i t y f a c t o r s o f the CR c y c l o t r o n  117  XVIII.  Measured  frequency s h i f t s  caused by t u n i n g b e l l o w s  121  Measured frequency s h i f t s deflection  caused by ground arm t i p  XIX.  121 vi  Measured and computed resonances f o r a one Dee-resonant l i n e system Measured and computed resonances f o r a two Dee-resonant l i n e system  vii  LIST OF FIGURES  Page 19.  Schematic o f the t u n i n g b e l l o w s  156  20.  Percentage frequency change vs p o s i t i o n o f t u n i n g b e l l o w s (CRM) a) the fundamental b) the t h i r d harmonic; c a l c u l a t i o n based on p e r t u r b a t i o n t h e o r y by S l a t e r  157  Percentage frequency change vs p o s i t i o n o f diaphragm i n the r e s o n a t o r c a l c u l a t e d u s i n g p e r t u r b a t i o n theory by S l a t e r (zero p o s i t i o n r e f e r s t o the r o o t ; the diaphragm i s i n i t s vertical position)  158  Percentage frequency change vs p o s i t i o n o f diaphragm i n the resonator; c a l c u l a t i o n based on a change o f the t o t a l c a p a c i t a n c e o f the r e s o n a t o r (the diaphragm i s i n i t s vertical position)  159  Extreme end s e c t i o n a) top view b) s e c t i o n w i t h m o d i f i e d hot arms  160  C e n t r a l r e s o n a t o r segments a) top view b) s i d e view c) g r i d o f p o i n t s f o r v o l t a g e measurements  161  Beam-RF f i e l d i n t e r a c t i o n a) g e n e r a t o r and a beam c u r r e n t beam p u l s e c) r e p r e s e n t a t i o n l o a d d) beam induced v o l t a g e  162  21.  22.  23.  24.  25.  26.  27.  28.  r e s o n a t o r f e d by an RF g e n e r a t o r b) amplitude o f a o f a r e s o n a t o r w i t h a beam and t o t a l v o l t a g e on r e s o n a t o r  Power d e l i v e r e d t o the beam a) and frequency change b) vs i n j e c t i o n phase (the RF fundamental, I„ = 100 yA, E„ = 500 MeV) B B Power d e l i v e r e d t o the beam a) and frequency change b) vs i n j e c t i o n phase (the t h i r d harmonic, I = 100 yA, E^ = 500 MeV) B Power d e l i v e r e d t o the beam a) and frequency change b) v s i n j e c t i o n phase (the RF fundamental, I = 750 yA, E = 400 MeV) c  29.  30.  Power d e l i v e r e d to the beam a) and frequency change b) v s i n j e c t i o n phase (the t h i r d harmonic, I = 750 yA, E,, = 400 MeV) B Power d e l i v e r e d t o the beam v s beam phase width a) the RF fundamental b) the t h i r d harmonic ( I = 100 yA, E_ = 500 MeV) B B  31.  32.  33.  163  164  165  166  167  Frequency change vs beam phase w i d t h ( 1 ^ = 1 0 0 yA, Eg = 500 MeV) a) the RF fundamental b) the t h i r d harmonic  168  T o t a l RF fundamental v o l t a g e on r e s o n a t o r d u r i n g t h e i n j e c t i o n of a beam (without the RF t h i r d harmonic)  169  T o t a l RF fundamental v o l t a g e on r e s o n a t o r d u r i n g the i n j e c t i o n o f a beam (with t h e RF t h i r d harmonic) ix  170  Page 34.  T o t a l RF t h i r d harmonic v o l t a g e on r e s o n a t o r d u r i n g t h e i n j e c t i o n o f a beam (with t h e RF t h i r d harmonic)  171  Components o f r e s o n a t o r v o l t a g e d u r i n g the course o f a c c e l e r a t i o n a) RF v o l t a g e amplitudes due to e x t e r n a l sources b) the f i r s t harmonic component of beam induced v o l t a g e (RF fundamental o p e r a t i o n )  172  Components o f r e s o n a t o r v o l t a g e d u r i n g t h e course of a c c e l e r a t i o n (RF f l a t - t o p o p e r a t i o n ) a) t h e f i r s t harmonic component o f beam induced v o l t a g e b) t h e t h i r d harmonic component o f beam induced v o l t a g e  173  37.  C o u p l i n g loop assembly  174  38.  R e p r e s e n t a t i o n o f c o u p l i n g a) and lumped parameter r e p r e s e n t a t i o n o f a Dee resonant system b)  175  35.  36.  39.  Input impedance vs d e t u n i n g from resonance  a) magnitude o f  i n p u t impedance b) phase o f i n p u t impedance  176  40.  Resonator  177  41.  R e p r e s e n t a t i o n o f t h e system w i t h d i s t r i b u t e d (program RESLINE)  42.  Matching u s i n g a nA/2 l i n e a) r e p r e s e n t a t i o n o f a Dee w i t h lumped parameters and a matching network b) Dee and l i n e i n t h e v i c i n i t y o f f c) impedance t r a n s f o r m a t i o n u s i n g TT network  179  R e p r e s e n t a t i o n o f t h e system a)two Dees r e p r e s e n t e d w i t h lumped parameters and t h e t r a n s m i s s i o n l i n e w i t h d i s t r i b u t e d parameters (program MATCH) b) lumped parameter r e p r e s e n t a t i o n o f t h e whole system  180  Phase s h i f t between loop and r e s o n a t o r r o o t c u r r e n t s v s d r i v i n g frequency  181  peak v o l t a g e vs d e t u n i n g from resonance parameters  178  Q  43.  44. 45.  Resonator t i p to loop v o l t a g e r a t i o v s d r i v i n g a) magnitude o f V,/V" b) phase o f V,/ V"  46.  Percentage frequency change v s stub s h o r t i n g p l u n g e r p o s i t i o n a) 4 stubs coupled c a p a c i t i v e l y t o a Dee made up o f 10 s e c t i o n s b) 4 stubs coupled c a p a c i t i v e l y to a two Dee r e s o n a t o r , each Dee made up o f 10 s e c t i o n s  183  V o l t a g e v a r i a t i o n along the a c c e l e r a t i n g gap vs stub s h o r t i n g p l u n g e r p o s i t i o n a) 4 stubs coupled c a p a c i t i v e l y t o a Dee made up o f 10 s e c t i o n s b) 4 stubs coupled c a p a c i t i v e l y to a two Dee r e s o n a t o r , each Dee made up o f 10 s e c t i o n s  184  47.  48.  frequency  Percentage frequency change and q u a l i t y f a c t o r v a r i a t i o n v s stub s h o r t i n g p l u n g e r p o s i t i o n (a stub coupled c a p a c i t i v e l y to a Dee made up o f 5 s e c t i o n s ) x  185  V o l t a g e v a r i a t i o n along the h o t arm t i p s vs ground arm t i p d e f l e c t i o n (Dee made up o f 5 s e c t i o n s ) a) the fundamental b) the t h i r d harmonic Frequency t u n i n g by means o f c y l i n d r i c a l c a p a c i t o r s (Dee made up o f 10 s e c t i o n s ) a) percentage frequency change and b ) q u a l i t y f a c t o r v a r i a t i o n v s p o s i t i o n of c y l i n d r i c a l capacitors V o l t a g e v a r i a t i o n a l o n g the h o t arm t i p s v s p o s i t i o n o f cylindrical capacitors Q u a l i t y f a c t o r v a r i a t i o n a) and v o l t a g e v a r i a t i o n along the h o t arm t i p s b) vs p o s i t i o n o f c a p a c i t i v e t u n i n g p l a t e s (Dee made up o f 10 s e c t i o n s ) Frequency t u n i n g by means o f r o t a t i n g f i n s (measurements done w i t h a 3 s e c t i o n r e s o n a t o r w i t h the f i n s i n s e r t e d i n the upper and lower c e n t r e segments) a) percentage frequency change and b) q u a l i t y f a c t o r v a r i a t i o n v s p o s i t i o n o f tuning f i n s Frequency t u n i n g by means o f r o t a t i n g f i n s (measurements done w i t h a two s e c t i o n r e s o n a t o r w i t h 8 f i n s p e r each segment) a) percentage frequency change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n of f i n s Frequency t u n i n g by means o f r o t a t i n g loops (measurements done w i t h a two s e c t i o n r e s o n a t o r w i t h 8 l o o p s p e r each segment) a) percentage frequency change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n of loops Resonator frequency t u n i n g by means o f diaphragms a) schematic o f the e x p e r i m e n t a l arrangement b) r e s o n a t o r w i t h t u n i n g diaphragms Frequency t u n i n g by means o f t u n i n g diaphragms a) percentage frequency change and b) t h i r d - t o - f i r s t harmonic frequency r a t i o vs p o s i t i o n o f diaphragms; p o s i t i o n o f diaphragms i n r e s o n a t o r a t 108.5 i n . from r o o t Frequency t u n i n g by means of t u n i n g diaphragms a) percentage frequency change and b) t h i r d - t o - f i r s t harmonic frequency r a t i o vs p o s i t i o n o f diaphragms; p o s i t i o n o f diaphragms i n r e s o n a t o r a t 65.4 i n . and 21.65 i n . from r o o t Resonator frequency t u n i n g by means o f a t u n i n g stub a) e x p e r i m e n t a l arrangement b) lumped constant r e p r e s e n t a t i o n Percentage v o l t a g e change v s h o t arm d e f l e c t i o n a) d e f l e c t i o n of the upper hot arm #3 i n an 18 s e c t i o n one Dee r e s o n a t o r b) d e f l e c t i o n o f the lower h o t arm #6B i n a 20 s e c t i o n two Dee r e s o n a t o r  Page 61.  Percentage frequency change vs hot arm d e f l e c t i o n a) d e f l e c t i o n o f the upper hot arm #3 i n an 18 s e c t i o n one Dee r e s o n a t o r b) d e f l e c t i o n of the lower hot arm #6B i n a 20 s e c t i o n two Dee r e s o n a t o r  198  Percentage v o l t a g e change vs r o o t p l u n g e r p o s i t i o n a) motion of r o o t p l u n g e r s i n s e c t i o n #7 i n ah 18 s e c t i o n one Dee r e s o n a t o r b) motion of r o o t p l u n g e r s i n s e c t i o n #7A i n a 20 s e c t i o n two Dee r e s o n a t o r  199  Percentage frequency change vs r o o t plunger p o s i t i o n a) motion of r o o t p l u n g e r s i n s e c t i o n #7 i n an 18 s e c t i o n one Dee r e s o n a t o r b) motion of r o o t p l u n g e r s i n s e c t i o n #7A i n a 20 s e c t i o n two Dee r e s o n a t o r  200  Percentage frequency change a) and q u a l i t y f a c t o r v a r i a t i o n b) vs motion of r o o t p l u n g e r s i n s e c t i o n #4 i n a 5 s e c t i o n one Dee r e s o n a t o r  201  65.  C e n t r a l r e g i o n geometry  202  66.  V o l t a g e d i s t r i b u t i o n a l o n g the t r a n s m i s s i o n l i n e ; b a s i c set-up  203  V o l t a g e d i s t r i b u t i o n along the t r a n s m i s s i o n magnitude of CP(3) changed to 415 pF  line; 204  V o l t a g e d i s t r i b u t i o n a l o n g the t r a n s m i s s i o n  line;  62.  63.  64.  67.  68.  p o s i t i o n of CP(3)  changed to 327  cm  205  69.  V o l t a g e phase a l o n g the t r a n s m i s s i o n l i n e ; b a s i c set-up  206  70.  CRM  207  71.  RF c o n t a c t s  one  Dee  resonator  208  xxx  ACKNOWLEDGEMENTS  I would l i k e  to thank Dr. K.L. Erdman f o r s u p e r v i s i n g  t h i s work and f o r p r o v i d i n g guidance and h e l p f u l s u g g e s t i o n s throughout the course o f my s t u d i e s a t U.B.C. l i k e t o thank Mr. O.K. F r e d r i k s s o n  I would  also  f o r many h e l p f u l d i s c u s -  s i o n s and the members o f the TRIUMF RF group f o r t h e i r invaluable help.  F i n a l l y I would l i k e  to thank my w i f e  J i t k a f o r drawing the f i g u r e s . F i n a n c i a l support from the TRIUMF p r o j e c t throughout the course o f t h i s work i s g r a t e f u l l y acknowledged.  xiii  CHAPTER I . 1.  INTRODUCTION  TRIUMF CYCLOTRON IN GENERAL T r i - U n i v e r s i t y - M e s o n - F a c i l i t y i s an i s o c h r o n o u s s e c t o r - f o c u s e d ion  cyclotron,  500  MeV.  magnet.  designed to a c c e l e r a t e 100  Azimuthally-varying 2  must be of H  1  The  l i f e t i m e of H  as low  ions.  as 5.76  As  kG  —  magnetic f i e l d  i o n s to an energy of  i s provided  by  at 500  MeV  to 308  i n . and  r-,  i o n s i s produced e x t e r n a l l y i n a hot The  f i l a m e n t Penning  beam i s a c c e l e r a t e d to 300  keV  and  arc  is  a spiral electro-  3  mflector.  The  RF  system c o n s i s t s o f two  p u s h - p u l l mode.  The  frequency.  Since  each t u r n .  The  RF  MeV  the i o n r o t a t i o n frequency  i n j e c t e d a x i a l l y i n t o the median p l a n e of the c y c l o t r o n by  The  the 500  MHz.  s o u r c e of the E h l e r ' s type.  static  field  r a d i u s to prevent e x c e s s i v e d i s s o c i a t i o n  a consequence of a low maximum magnetic f i e l d ,  The beam of H  .  a six-sector  i o n s r e q u i r e s that the maximum magnetic  o r b i t radius i s approximately equal i s o n l y 4.62  yA of H  negative  RF  180  deg wide Dees o p e r a t i n g  i n the  frequency i s the f i f t h harmonic of the i o n r o t a t i o n  the Dee  voltage  i s 100  kV,  beam must thus complete 1250  t r a n s m i t t e r must be  the p a r t i c l e s g a i n 400 turns before  capable of d e l i v e r i n g up  d i s s i p a t e d i n the r e s i s t i v e l o s s e s and  to 1.5  keV  on  i t reaches 500 MW  (cw)  MeV.  of RF power  the power s u p p l i e d to the beam.  To keep the r a d i a t i o n l e v e l below p e r m i s s i b l e l i m i t s a v e r y  good vacuum  -7 of the o r d e r  of 10  T o r r must e x i s t .  E x t r a c t i o n i s achieved passing  by  s t r i p p i n g b o t h e l e c t r o n s on the i o n s  the beam through a t h i n f o i l ,  o r b i t i n the magnetic f i e l d . distances  thus r e v e r s i n g the c u r v a t u r e  By p o s i t i o n i n g the s t r i p p i n g f o i l  by of  at  the  different  from the c e n t r e of the c y c l o t r o n , the output energy v a r i e s between it  200  MeV  and  500  MeV.  The  beam i s thus e x t r a c t e d w i t h n e a r l y 100%  Another advantage of t h i s e x t r a c t i o n mechanism i s a p o s s i b i l i t y of  efficiency. simul-  - 2 taneous e x t r a c t i o n o f s e v e r a l p r o t o n beams a t d i f f e r e n t The  energies.  c y c l o t r o n w i l l produce mesons which can be used i n examining t h e  s t r u c t u r e o f t h e mesons themselves, i n photographing atomic n u c l e i w i t h meson beams, and i n forming  muonic and p i o n i c atoms.  operation are envisaged.  Since  Two b a s i c modes  5  of cyclotron  the mesons a r e produced i n secondary t a r g e t s  the c u r r e n t must be as l a r g e as 200 uA, b u t the energy r e s o l u t i o n w i l l be low, AE = ±600 keV. possible.  A beam w i t h  produced d u r i n g acceleration. and,  5  r e q u i r e as h i g h a duty c y c l e as  a h i g h energy r e s o l u t i o n o f AE = ±50 keV w i l l be  the o t h e r mode o f c y c l o t r o n o p e r a t i o n , i . e . s e p a r a t e d I n t h i s case,  consequently,  harmonic helps  T h i s mode o f o p e r a t i o n w i l l  the m i c r o s c o p i c  turn  duty f a c t o r w i l l be v e r y low  t h e beam c u r r e n t w i l l be o n l y 20 yA.  Adding t h e t h i r d  o f the RF to the RF fundamental f l a t - t o p s t h e RF v o l t a g e wave and  to e i t h e r i n c r e a s e the phase acceptance g i v i n g a l a r g e r beam c u r r e n t  or t o m a i n t a i n  s p a t i a l t u r n s e p a r a t i o n out to e x t r a c t i o n .  - 3 -  2.  TRIUMF CYCLOTRON RF SYSTEM The  TRIUMF RF system possesses  a p p l i e d anywhere e l s e .  unique f e a t u r e s t h a t have not been  The r e s o n a t o r i s made up o f two Dees subtending  angle o f 180 deg a t the a c c e l e r a t i n g gap. c o n s i s t s o f two quarter-wave l o n g , s h o r t e d  The Dee s t r u c t u r e i t s e l f transmission l i n e s  an  ( F i g . 1)  coupled  c a p a c i t i v e l y a t the h i g h v o l t a g e end, which r e s u l t s i n two modes o f p o s s i b l e operation.  Only  one o f them, namely a p u s h - p u l l mode, i s u s e f u l f o r  a c c e l e r a t i o n of i o n s .  The r e s o n a t o r i s d i v i d e d i n t o segments ( F i g . 2) i n  o r d e r to s i m p l i f y manufacturing  and h a n d l i n g and t o a v o i d p a r a s i t i c modes.  Each Dee i s made up o f two rows (upper 20 r e s o n a t o r segments ( F i g . 3 ) . thus  and l o w e r ) , each o f them c o n s i s t i n g o f  The r e s o n a t o r p a n e l s a r e c a n t i l e v e r e d , and  the t r o u b l e s f r e q u e n t l y e x p e r i e n c e d  i f i n s u l a t o r s a r e used can be  avoided. In the TRIUMF c y c l o t r o n the beam o f 100 yA o f H~ i o n s w i l l be accelerated H  t o the output  i o n s i n a magnetic f i e l d  orbit  to about 5.76 kG.  and  an u n a c c e p t a b l e  ion  o r b i t i n g frequency  frequency and  energy o f 500 MeV.  the maximum magnetic f i e l d  radiation.  of  a t the 500 MeV  T h i s l i m i t on the magnetic f i e l d p l a c e s the  a t approximately  4.62 MHz, which i s r a t h e r a low  as compared t o o t h e r c y c l o t r o n s .  To reduce the s i z e o f the Dees  the power r e q u i r e d the RF system w i l l o p e r a t e  The  7  A h i g h e r f i e l d would b r i n g about h i g h e r beam l o s s e s  the i o n o r b i t i n g frequency  a t the f i f t h harmonic o f  (the RF power i s thus reduced  by a f a c t o r o f / 5 ) .  c y c l o t r o n performance may be improved by f l a t - t o p p i n g t h e RF v o l t a g e  wave ( F i g . 4 ) . to  limits  The e l e c t r i c d i s s o c i a t i o n  The r e s o n a t o r d e s i g n a l l o w s the a d d i t i o n o f h i g h e r harmonics  the fundamental c a v i t y mode.  would r e q u i r e s p e c i a l treatment i n c r e a s e the mechanical  However, adding h i g h e r harmonics than h = 3 o f the r e s o n a t o r s u r f a c e s and would s e v e r e l y  t o l e r a n c e s r e q u i r e d i n o r d e r to a t t a i n h i g h  f a c t o r s f o r these h i g h e r harmonics.  quality  A l s o , the c y c l o t r o n o p e r a t i o n i s n o t  - 4 much improved by e x c i t i n g r e s o n a t o r modes w i t h h > 5. y  F o r these reasons  only  the t h i r d harmonic o f the fundamental, f o r which the RF c a v i t y changes from a s h o r t e d A/4 l i n e to a 3/4A l i n e , w i l l be used.  Adding about 11% o f the  t h i r d harmonic o f the RF to the fundamental mode i n c r e a s e s the m i c r o s c o p i c duty f a c t o r and makes p o s s i b l e s e p a r a t e d The  turn acceleration.  c h o i c e o f the a c c e l e r a t i n g v o l t a g e i s determined  power as w e l l as by the s p a r k i n g c r i t e r i o n . power tubes  by the a v a i l a b l e RF  T a k i n g i n t o c o n s i d e r a t i o n the  a v a i l a b l e a t t h e time o f the TRIUMF p r o p o s a l , the a c c e l e r a t i n g  v o l t a g e was s e t a t 200 kV. The by  t o l e r a n c e s on the RF v o l t a g e and frequency  the d e s i r e d q u a l i t y o f the outcoming beam.  and v o l t a g e amplitude  s t a b i l i t y are determined  Both the r e s o n a t o r  w i l l be a u t o m a t i c a l l y c o n t r o l l e d .  frequency  The frequency  will  be h e l d constant by means o f t u n i n g b e l l o w s mounted a t the r e s o n a t o r r o o t . F a s t v o l t a g e amplitude the f i n a l t e t r o d e stage  c o n t r o l w i l l be achieved  through  s c r e e n modulation o f  (Fig. 5).  A x i a l i n j e c t i o n o f i o n s w i l l be used i n the TRIUMF c y c l o t r o n . A c c o r d i n g ly,  the r e s o n a t o r segments i n the c e n t r a l r e g i o n o f the c y c l o t r o n w i l l be  m o d i f i e d t o a l l o w the i n s t a l l a t i o n A resonant to  o f the c e n t r e p o s t .  t r a n s m i s s i o n l i n e and a loop f o r c o u p l i n g a r e b e i n g  t r a n s f e r power from the main power a m p l i f i e r t o the Dees.  employed  The t h i r d  harmonic mode w i l l be e x c i t e d by means o f a s e p a r a t e l i n e and c o u p l i n g l o o p .  3.  CRITICAL RF I t was  PARAMETERS necessary  to prove t h a t the r e s o n a t o r segments c o u l d be a l i g n e d  m e c h a n i c a l l y so as to not a f f e c t system, i . e . m a i n l y a c c e l e r a t i n g gap,  the e l e c t r i c a l p r o p e r t i e s o f the  the s t a b i l i t y of the resonant lower  Another q u e s t i o n was  the v o l t a g e u n i f o r m i t y .  mechanical  oscillations  frequency,  and  how  i n o r d e r not  to lower  the q u a l i t y  can be expected.  to show t h a t the frequency  could occur.  I t was  Should  they produce a l a r g e f r e q u e n -  l a r g e so t h a t no f l i p p i n g  steady  s t a t e w i t h a good phase and  and  p o s s i b l e to  i n t o the r e s o n a t o r and y e t a c h i e v e  frequency  a  stability.  r e s o n a t o r t i p l o a d i n g c a p a c i t y as r e q u i r e d by the c o n d i t i o n of  resonance i s not  the same f o r both harmonics.  harmonic resonant  frequency  i s g e n e r a l l y not  f i r s t harmonic resonant  frequency.  more than the o t h e r had  to be found.  f o r both  of the RF  effect.  from one mode i n t o  a l s o not obvious whether i t was  the two harmonics s i m u l t a n e o u s l y  gated  factors  s e p a r a t i o n between the p u s h - p u l l  inject  The  tips  S i n c e no i n s u l a t o r s are b e i n g used, some  push-push modes i s s u f f i c i e n t l y another  the phase r e l a -  good the c o n t a c t between the hot arm  cy v a r i a t i o n , dampers would have to be used to minimize t h e i r We had  the  rows o f the r e s o n a t o r segments and between  and between the r o o t p i e c e s must be and  resonator  the q u a l i t y f a c t o r s , the v o l t a g e u n i f o r m i t y a l o n g  t i o n between the upper and the Dees.  5 -  the f i r s t  and  For t h i s r e a s o n three times  the  third  as l a r g e as  A t u n i n g element i n f l u e n c i n g one The  frequency  t u n i n g had  the t h i r d harmonic of the RF.  to be  The  the harmonic investi-  exact  value  fundamental frequency w i l l not be known u n t i l the main magnet  are completed.  The  tests  r e s o n a t o r must, t h e r e f o r e , be p r o v i d e d w i t h some k i n d of  t u n i n g i n o r d e r to a c h i e v e i s o c h r o n i s m . i n v e s t i g a t e d - f i n e t u n i n g and designed which would not  Two  k i n d s of t u n i n g were to be  coarse t u n i n g .  Such a method had  to be  cause the v o l t a g e v a r i a t i o n along the main gap  i n c r e a s e to a v a l u e g r e a t e r than a c c e p t a b l e  limits.  to  - 6 In the main c y c l o t r o n the beam l o a d i s expected t o t a l amount of the RF power.  to be up to 1/4 o f the  I f we a l l o w such a beam of charged p a r t i c l e s t o  pass through a r e s o n a t o r , a l l p o s s i b l e harmonic modes of the fundamental c a v i t y frequency operation.  can be e x c i t e d , some o f them u n d e s i r a b l e f o r t h e RF system  The induced  a c c e l e r a t i n g gap.  field  g i v e s r i s e t o a p e r i o d i c v o l t a g e V_ a c r o s s the  Moreover, i f t h e r e i s an RF f i e l d  passage o f the beam, we a l s o get a beam-RF f i e l d  i n the c a v i t y d u r i n g the  i n t e r a c t i o n which r e s u l t s  e i t h e r i n a d e l i v e r y o r i n an a b s o r p t i o n o f the energy i n the e x i s t i n g RF field.  I n the case o f the f i r s t harmonic o f the RF, i t means t h a t the beam  absorbs a c e r t a i n amount o f the RF energy.  However, when both  t h e f i r s t and  the t h i r d harmonic of the RF are on, the beam may couple some amount o f energy from the f i r s t i n t o the t h i r d harmonic. e f f e c t s , the beam can a l s o cause a d e t u n i n g  Beside  the two mentioned  o f the r e s o n a t o r s h o u l d t h e beam  p u l s e s n o t appear as a r e s i s t i v e l o a d (zero phase a n g l e ) . One o t h e r phenomenon - namely, m u l t i p a c t o r i n g - i s a s s o c i a t e d w i t h onator h i g h v o l t a g e o p e r a t i o n under vacuum.  res-  The m u l t i p a c t o r i n g c u r r e n t due t o  o s c i l l a t i n g e l e c t r o n s i n c r e a s e s v e r y r a p i d l y because o f e l e c t r o n m u l t i p l i c a t i o n on the c a v i t y w a l l s . and  T h i s c u r r e n t produces a heavy l o a d on the RF g e n e r a t o r ,  i t i s o f t e n i m p o s s i b l e t o reach  a l s o l e a d t o sparkovers  the o p e r a t i n g v o l t a g e .  because o f i o n i z e d r e s i d u a l gas.  The non-uniform shape o f the r e s o n a t o r s due  t o a cut-out which i s n e c e s s a r y  m e c h a n i c a l s t a b i l i t y o f t h e magnet. comparable w i t h  M u l t i p a c t o r i n g can  a t the c e n t r e o f the machine i s  t o accommodate a c e n t r e post r e q u i r e d f o r S i n c e the dimensions o f the c u t - o u t a r e  the q u a r t e r wavelength o f t h e t h i r d harmonic o f the RF, the  e l e c t r i c a l p r o p e r t i e s - the Q and v o l t a g e u n i f o r m i t y along the Dee gap - c o u l d be damaged by an improper shaping  o f the r e s o n a t o r segments.  The o r i g i n a l  p r o p o s a l of the r e s o n a t o r system a l s o c a l l e d f o r m o d i f i e d extreme end segments. Owing t o a c i r c u l a r shape o f t h e vacuum chamber, t h e segments were t o be  - 7 tapered t o f i t the vacuum chamber. quality  the t h i r d harmonic  factor.  Some i n v e s t i g a t i o n s had resonant  This could again a f f e c t  to be made as to whether a resonant  l i n e s h o u l d be used to t r a n s f e r the power to the Dees.  or  non-  By a proper  c h o i c e of the parameters a v e r y h i g h s t a n d i n g wave r a t i o can e x i s t i n the resonant  line.  With a h i g h s t a n d i n g wave r a t i o i n the l i n e the system i s l e s s  s e n s i t i v e to v a r y i n g l o a d s , i . e . a sparkover  or a r e a c t i v e beam l o a d g i v i n g  r i s e to a mismatch i s e a s i e r to c o n t r o l .  the o t h e r hand, the main d i s -  On  advantage i s i n the i n c r e a s e d s i z e o f the l i n e . resonant  l i n e must be  chosen to prevent  The  dimensions f o r the  s p a r k i n g and y e t to t r a n s f e r  the  r e q u i r e d power to the Dees. Computer programs had  to be w r i t t e n to f i n d  the l i n e parameters f o r a  matched system, t o a s c e r t a i n the e f f e c t of a changed frequency t r a n s m i s s i o n l i n e parameters, to f i n d how beam l o a d , and  e x c i t e both Dees.  The  of the whole RF system.  whether c o u p l i n g to one Dee  A l l the mentioned problems had  f i n a l d e s i g n of the r e s o n a t o r c o u l d be  the  the system i s s e n s i t i v e to a v a r y i n g  to i n v e s t i g a t e the resonances  q u e s t i o n to answer was  on  only i s s u f f i c i e n t  finished.  by the d e s i r e d q u a l i t y of the beam,  5  to  to be looked i n t o b e f o r e  t o l e r a n c e s on the RF parameters, summarized i n T a b l e I, are  determined  The  Table I I .  the  - 8 -  TABLE I Tolerances.on.the  RF parameters  Maximum duty factor  Frequency  stability  ± 1.25/10  Voltage s t a b i l i t y (first) Phase 3 to 1 (deg) r d  Single turn extraction  fc  ± 2/10  L  ±1.5  s t  Voltage s t a b i l i t y (third) V o l t a g e asymmetry (central region)  ± 7.5/10  B  ± 2.5/10  5  ± 0.15 ±  5/10*  1/10 5/10  3  3  TABLE I I Beam F u l l width energy spread (keV)  Raw beam  ± 600  characteristics Spread i n phase (deg)  ± 12 ± 36 ( 3  Low energy s l i t s 0.048 i n . 0.032 i n .  Separated t u r n acceleration  Estimated intensity (VA)  r d  )  Duty  factor (%)  200  7  200  20  ± 220  ±  2  1  1.1  ± 140  ±  1.8  0.5  1.0  ± 140  ± 14 ( 3  ±  80  ±  ±  45  ±6.7  r d  )  0.5  ± 105  ±  , (3 ) 6.7  ±  ±  0.5  16  8.0  0.1  0.3  1  3.7  4  3.7  0.05  0.3  0.5  2.8  r d  Final orbit selection  60  ±5.0  , (3 ) r d  -  CHAPTER I I .  9  -  THEORETICAL CALCULATIONS  RESONATOR BASIC PARAMETERS AND RELATIONS 1.1  Characteristic  Impedance  I f a low s t a n d i n g wave r a t i o e x i s t s i n t h e t r a n s m i s s i o n l i n e power t o t h e Dees, an a c c u r a t e v a l u e o f  transferring  ( t h e c h a r a c t e r i s t i c impedance o f  the r e s o n a t o r r e p r e s e n t e d as a t r a n s m i s s i o n l i n e ) i s n e c e s s a r y f o r s e t t i n g t h e r e s o n a n t l i n e parameters.  A t i g h t c o u p l i n g between t h e top and b o t t o m rows  o f e i t h e r Dee i s p r o v i d e d by means o f f l u x guides  (see F i g . 1 ) .  t h e s e f l u x g u i d e s , w i t h a 4 i n . gap, make c a l c u l a t i o n o f  However,  more d i f f i c u l t .  T h e i r i n f l u e n c e on t h e c h a r a c t e r i s t i c impedance i s reduced as t h e number o f r e s o n a t o r segments i n c r e a s e s . T r e a t i n g t h e r e s o n a t o r as a s h o r t ~ c i r c u i t e d t r a n s m i s s i o n l i n e , t h e c h a r a c t e r i s t i c impedance i s g i v e n by  = A : + i^:  Z  c • v G  a.D  + jwC  I n a t r a n s m i s s i o n l i n e i n a normal u s e , G' fy 0,  Z  = Z  c with  Y  =  a  + j3, a = —  - jZ  o  J  and i t f o l l o w s  (1.2)  f  o 3  , 3 = - , Z o  o  = ^  .  R', L % G', C" a r e d i s t r i b u t e d parameters o f t h e r e s o n a t o r a c t i n g as a t r a n s m i s s i o n l i n e , c i s the v e l o c i t y of l i g h t , f = value of a, the a t t e n u a t i o n constant  i s the frequency.  (Appendix A ) , i s v e r y s m a l l , and f o r t h i s  r e a s o n we s h a l l assume Z  = Z unless s p e c i f i e d otherwise. c o c a p a c i t a n c e p e r u n i t l e n g t h may be w r i t t e n as C' = 8.854 e  r  g  xio"  The  Since the  1 2  where w i s the w i d t h of t h e r e s o n a t o r , g i s t h e r e s o n a t o r gap, e  i s the  dielectric  c o n s t a n t o f medium.  10 -  The MKSA system of u n i t s i s used i n a l l  c a l c u l a t i o n s unless s p e c i f i e d otherwise.  The  c h a r a c t e r i s t i c impedance of  r e s o n a t o r c o n s i s t i n g of n segments i s expressed  Z  where w = w  a  .n , w  ly different  1.2  = 120TT  o  as  (1.3)  K  w  i s the average width  from the nominal width  T i p Loading Although  a  of r e s o n a t o r segment,  use  w  a  i s general-  of a segment.  Capacity  the approximate v a l u e of t h i s c a p a c i t a n c e can be  u s i n g s e v e r a l d i f f e r e n t methods, the exact v a l u e w i l l we  the  calculated  always be  a more p r e c i s e c a l c u l a t i o n to o b t a i n the c o r r e c t f i e l d  by  unknown u n l e s s  and  potential  lines. S i n c e the r e s o n a t o r gap e f f e c t on the hot arm arm  i s q u i t e s m a l l , the ground arms w i l l have some  tip-to-tip  capacity.  The  field  t i p s i n the r e a l r e s o n a t o r d e s i g n would d i f f e r  l i n e s between two  hot  from the arrangement where  the two h o t arms are p l a c e d o p p o s i t e each o t h e r i n a f r e e space.  Simultaneous-  l y , other mechanical  affect  n o n - u n i f o r m i t i e s , such as beam probe h o u s i n g ,  the  true value. An o v e r - e s t i m a t e  of the t i p - t o - t i p  assuming t h a t the two hot an e x p r e s s i o n r e s u l t i n g i n t o account  c o u p l i n g c a p a c i t y i s .made by  arms are p l a c e d i n a f r e e space.  gives  from the conformal mapping t r a n s f o r m a t i o n which  a t r a n s m i s s i o n l i n e c o n s i s t i n g of two  R e p l a c i n g the two  Literature  hot arm  t i p s by two  parallel  c i r c u l a r conductors  as t h a t of a t i p , the c h a r a c t e r i s t i c impedance of t h i s  circular  takes  conductors.  of the same r a d i u s  two-conductor l i n e i s  g i v e n by  (1.4) /D/2p / D/2p  + 1 - 1  - 11 where D i s the d i s t a n c e between the c e n t r e s o f t h e conductors r a d i u s o f conductor. Z = 275 ^/m.  S u b s t i t u t i n g D = 7.5 i n . and p = 0.75 i n . r e s u l t s i n  The c a p a c i t a n c e p e r u n i t l e n g t h i s then C = 12.1 pF/m.  f i x e d h o t arm width arm  = 19.68 pF.  r e s u l t e d i n C^,_^ = 7.5 pF p e r resonant  resonant lower  The measurements at h a l f s c a l e have  s e c t i o n , which means C^,_^, = 15 pF a t  From now on we w i l l denote a t i p - t o — t i p c a p a c i t a n c e between two  s e c t i o n s as 0^,  .  r e s o n a t o r segments.  A resonant  s e c t i o n i n c l u d e s both  the upper and  A h o t arm t i p - t o - g r o u n d p l a n e c a p a c i t a n c e p e r one  segment, which i s t h e same i n magnitude as  1.3  For a  o f 32 i n . , and c o n s i d e r i n g both t h e upper and lower h o t  t i p s , we a r r i v e a t  f u l l scale.  and p i s t h e  w i l l be denoted as C ^ p .  Resonant Frequency F o r g i v e n v a l u e s o f t h e r e s o n a t o r c h a r a c t e r i s t i c impedance Z , r e s o n a t o r q  l e n g t h &, and t h e t i p - t o - g r o u n d l o a d i n g c a p a c i t y C^-j-p'  t  r e s o n a t o r o s c i l l a t i n g i n a p u s h - p u l l mode i s determined ing transcendental  n  frequency  e  o f the  by s o l v i n g t h e f o l l o w -  equation:  I jZ  tan\\ c I  = l  1  ^ C Q  (  1  '  5  )  1 T 1 T  T h i s e x p r e s s i o n a c t u a l l y f i x e s a l l odd harmonic resonant  f r e q u e n c i e s , as w e l l .  The push-push mode f r e q u e n c i e s a r e c a l c u l a t e d from the c o n d i t i o n t h a t  -r  = k'-  c  (  1  ,  6  )  2  where k ' i s a p o s i t i v e i n t e g e r and l* i s t h e d i s t a n c e between the r o o t and the centre nodal The if  plane.  equations  above a r e v a l i d  f o r a lossless transmission l i n e .  However,  the q u a l i t y f a c t o r i s s u f f i c i e n t l y h i g h , a t l e a s t o f the o r d e r o f magnitude  of 1000, the resonant  frequency  l o s s e s i n the d i e l e c t r i c .  o f the r e s o n a t o r i s almost  The frequency  u n a f f e c t e d by the  o f damped o s c i l l a t i o n s i s g i v e n by  - 12 /,  -  1  (1.7)  S i n c e the q u a l i t y f a c t o r i s of the o r d e r of magnitude of 7000, the r e s o n a t o r frequency unless s p e c i f i e d  by aj i s j u s t i f i e d .  From now  Q  on, co  approximating  w i l l be used  otherwise.  In the f i r s t  approximation  the resonant  frequency  depends on the  tip-to-  ground plane c a p a c i t y as  Afo _  "^TIP C  S i m i l a r l y , we  (1.8)  TIP 1 +  get  where s i n 2 ^ £  Afo _  AZ  fo  Z  0  (1.9)  0  1 + * sin  c  = s i n 2d), <f> b e i n g the f o r e s h o r t e n i n g angle.  r e s o n a t o r the f r a c t i o n i n eqn.  1.4  — s i n 2^o-£ c  (1.9)  i s l e s s than 1/15  For the TRIUMF  f o r the  fundamental.  Resonator Power Loss There are two major energy l o s s e s i n the r e s o n a t o r which must be s u p p l i e d  by  the e x t e r n a l d r i v i n g mechanism.  l o s s e s i n the r e s o n a t o r w a l l s to  the beam and  referred  The  first  (skin losses).  c o n s i s t s of the j o u l e h e a t i n g The  second i s the power s u p p l i e d  to the g i v e n beam c u r r e n t and energy.  Variations in  the beam c u r r e n t r e s u l t i n c o r r e s p o n d i n g v a r i a t i o n s i n the amount of power r e q u i r e d t o m a i n t a i n the v o l t a g e amplitudes specified level. to  i n the c y c l o t r o n r e s o n a t o r at  Moreover, a minor amount of power w i l l be p r o v i d e d i n o r d e r  compensate f o r the s k i n l o s s e s i n the resonant  the c o u p l i n g loop  the  t r a n s m i s s i o n l i n e and i n  (see S e c t i o n I I . 7 . 1 ) .  An e x p r e s s i o n f o r the r e s o n a t o r power l o s s due d e r i v e d from the t r a n s m i s s i o n l i n e  equations  to the s k i n e f f e c t i s  - 13 -  1 = V  T  V = Z I Q  sinh yx/Z + I Q  sinh yx  T  + V  T  cosh yx  (1.10)  cosh yx  (1.11)  which for a short c i r c u i t e d l i n e simplify to 1 = 1^ cosh Y  X  V = Z I sinh YX o T Since y = a + j g , a << 3 , t h e resonator power loss i s calculated from m  2  1  ll^j cos (-rx) dx 2  o  n |Vj " 2^6w^ a o  s i n 2-1 ( £  —>  +  ( 1  '  1 2 )  c where y i s the complex propagation constant, V^,I^, are the voltage and current at load (resonator root), |V | = Z ll^J i s the voltage peak i n the l i n e which i s a quarter wavelength long, R' =  i s the surface resistance (fi/m) a  for one panel, n i s the number of segments,a i s the s p e c i f i c conductivity, i / ^S x l 0 " i s the skin depth and Z^ i s the c h a r a c t e r i s t i c impedance 6 = 6.62/ xlO 2  of resonator segment.  The power loss i n the resonator o s c i l l a t i n g i n an odd  harmonic mode of the fundamental mode frequency may be evaluated according to this expression.  Choosing the root current I  T  = -j|l | T  results i n V  = [V |-  From the inspection of eqn. (1.12) i t follows that the skin losses i n the t h i r d harmonic mode are higher by about a factor of / J than the losses i n the fundamental mode, provided that a l l other parameters appearing i n eqn. are constant.  (1.12)  Due to the large size of the f u l l - s c a l e Dees, each of them  3m x 16m, a few measurements had to be done at h a l f - s c a l e .  For the tests the  dimensions of the f u l l - s c a l e resonator segments were scaled down by a factor of 2 i n order to obtain the same c h a r a c t e r i s t i c impedance as that considered for the main cyclotron. The power loss i n the resonator modelled  at h a l f -  - 14 s c a l e i s determined from  pCl/2)  Now, the  =  /2-pCD  s h o u l d the two Dees be assembled  power l o s s i s determined from eqn.  and o s c i l l a t e  i n a push-push mode  (1.12), where I =  X/4  2  ^ i Z !/ oao L -  = o, 44  P  9  vTw  (1.13)  z  a o  cu  where h = 1 f o r the f i r s t harmonic and h = 3 f o r the t h i r d  harmonic.  So f a r n o t h i n g has been s a i d about the l o s s e s i n r o o t p i e c e s . i n t e g r a t i o n i n eqn.  (1.12), u n f o r t u n a t e l y , does not i n c l u d e t h i s  The  s m a l l amount  of power l o s s which i n the case o f the TRIUMF r e s o n a t o r r e p r e s e n t s about 3% o f the  t o t a l RF power d e l i v e r e d to the Dees.  /f  2nt  P = 3.776  Z  w  C  •  vaa  a  i  of  cu  i s the s p e c i f i c  C  T I  1.5  (1.14)  cu i s the c u r r e n t through a r o o t  c o n d u c t i v i t y of copper.  the r e s o n a t o r power on v a r i o u s parameters.  parameters were: f p = 13 pF.  V  1  from  i2  | l|  where t i s the h e i g h t of a r o o t p i e c e , I and o"  T h i s power l o s s i s c a l c u l a t e d  F i g . 6 shows the  piece  dependence  The nominal v a l u e s of  =23.1 MHz, I = 3.0678 m, Z = 46 Q,, V . - V, = 100 kV, o o ol 1 i s the r e s o n a t o r t i p v o l t a g e (V = V i f (J) < 5 deg).  Energy S t o r e d i n Resonator The v o l t a g e and c u r r e n t a c r o s s the c a p a c i t a n c e i n the resonant c i r c u i t  are  90 deg out o f time phase.  alternately  T h i s i m p l i e s t h a t the t o t a l energy i s  s t o r e d i n the e l e c t r i c and magnetic f i e l d s .  The t o t a l  energy  s t o r e d i n a resonant c i r c u i t i s then found by d e t e r m i n i n g e i t h e r the energy s t o r e d i n the c i r c u i t c a p a c i t a n c e or i n d u c t a n c e . The e l e c t r i c and magnetic energy d e n s i t i e s at any p o i n t i n the r e s o n a t o r are  g i v e n by  - 15 -  = T C'|v(xj'  (1.15)  (1.16)  w.H where V ( x ) , I ( x ) a r e determined energy i s then c a l c u l a t e d  W E  by eqns. (1.10) and (1.11)  electric  from  C-|v(xj dx  = 2  The  2  V \ C, 2 TIP 1  +  1  I n t e g r a t i n g and making use o f eqn. (1.5), we o b t a i n  £  -  s i n 2—£ c  n  |v/  +  s i n 2—1 c  4coZ„  (1.17)  The magnetic energy i s c a l c u l a t e d i n a s i m i l a r way  W  H  = ^ 2  L ' | l ( x ) | dx s i n 2—£ c  n 1 4 cZ,  where we made use o f Z l l j = |V | and Z = cL". o T o o eqn.  (1.18)  Comparison o f eqn. (1.17) w i t h  (1.18) r e v e a l s t h a t the two e x p r e s s i o n s are i d e n t i c a l . I t a l s o f o l l o w s from eqn. (1.18) t h a t the s c a l i n g f a c t o r f o r going to a  h a l f - s c a l e model i s e q u a l to 1/2 because o n l y t h e r e s o n a t o r l e n g t h would be changed  (except the f r e q u e n c y ) .  oscillating  i n the t h i r d harmonic mode i s c a l c u l a t e d from eqn. (1.18), and i s  approximately that V  o  S i m i l a r l y the energy s t o r e d i n the r e s o n a t o r  e q u a l t o the energy s t o r e d i n the fundamental mode, p r o v i d e d  i s the same.  The average energy s t o r e d i n the r e s o n a t o r f o r v a r i o u s  v a l u e s o f r e s o n a t o r parameters can be e v a l u a t e d from F i g . 6. 1.6  Resonator Q u a l i t y F a c t o r As the q u a l i t y f a c t o r i n c r e a s e s the power which must be d e l i v e r e d to the  Dees d e c r e a s e s .  The h i g h q u a l i t y f a c t o r a l s o means t h a t a l l resonant  s e c t i o n s have been tuned  to the same resonant  frequency  and t h a t t h e r e i s t h e  - 16  -  u n i f o r m v o l t a g e d i s t r i b u t i o n along the main Dee f a c t o r s must not be itself, Q  confused, namely the  (unloaded q u a l i t y f a c t o r ) and  resonant system, Q^. expressed  The  gap.  Two  different quality  q u a l i t y f a c t o r of  the  resonator  the q u a l i t y f a c t o r of the whole  unloaded q u a l i t y f a c t o r of the  resonator i s  by W  OJ  where W i s the energy s t o r e d  i n the r e s o n a t o r i n resonance, P i s the power  oo l o s t i n the  r e s o n a t o r w a l l s and  s u b s t i t u t i o n f o r W and O  P  f  Q  = —  (root l o s s e s  Z_,CT<$W O  i s the  resonant f r e q u e n c y .  excluded) l e a d s  The  to  a  <- >  °-=—2~r— where w  i s the  f l u x guides).  1  average w i d t h of a segment The  parameters i n eqn.  (taking  into consideration  (1.19) have been d e f i n e d  t h i s f o r m u l a determines the q u a l i t y f a c t o r of the push-push modes, we  i s determined by  c o r r e s p o n d i n g resonant f r e q u e n c i e s of the  the  the  before.  resonator o s c i l l a t i n g  i n the p u s h - p u l l and  deduce that  the d i f f e r e n c e two  19  in  Since both Q's  modes.  In o r d e r to f i n d the s c a l i n g f a c t o r f o r a h a l f - s c a l e resonant c a v i t y , i t i s s u f f i c i e n t to take i n t o account the r e l a t i o n s d e r i v e d  i n previous  paragraphs:  Q  The  (l/2)  r a t i o between the  derived  1_ (D  =  Q  fundamental and  t h i r d harmonic q u a l i t y f a c t o r s  is  similarly:  Q  A more p r e c i s e  3  = /3  Q  1  c a l c u l a t i o n of the  computer program RESLINE.  The  q u a l i t y f a c t o r i s done by use  a d d i t i o n a l power l o s s i n the  root  of  the  shorting  - 17 plungers  and  -  i n the c o u p l i n g loop i s then  To measure the q u a l i t y f a c t o r two description follows.  The  considered.  d i f f e r e n t methods were used•  f i r s t method, the commonest i n microwave  i s based on measuring h a l f power p o i n t s (or .707 Assuming t h a t the r e s o n a t o r can be r e p r e s e n t e d R, L, C as a s e r i e s resonant  circuit,  Their circuits,  v o l t a g e peak p o i n t s ) .  i n terms of lumped parameters  the power absorbed  i n R i s g i v e n by  the  r e a l p a r t of  Iv I  2  P = R + At  jRQ ,C0  O  — co  (1.20)  J  resonance  Ivl  2  and  at  co =  co^  Re  (P)  and  co =  Re and  ='^4  co^  |v |  2  n  U  (P) =  the Q i s determined  from  co  Q =  The  —  Q  (1.21)  second method makes use of the damped o s c i l l a t i o n s of the f i e l d s i n  the r e s o n a t o r a c c o r d i n g to /  co  E(t) = E  Taking  e" -2$-  e  the a b s o l u t e v a l u e o f eqn.  Q  The  Q  =  1  ^  J 1  " ^  -  t  (1.22)  (1.22), the Q i s g i v e n by  2 l n ( E /E) o  r e l a t i o n s h i p between the Q's  7  ( 1  and  '  2 3 )  r e s o n a t o r parameters i s shown i n F i g . 7.  - 18 2.  RF TOLERANCES 2.1  Frequency Detuning Due to E v a c u a t i o n D u r i n g the e v a c u a t i o n  d e f l e c t inwards. height, for  Since  o f the Vacuum Chamber  o f t h e vacuum tank t h e s i d e s o f t h i s tank tend t o  the diameter o f the tank i s much l a r g e r than i t s  the d e f l e c t i o n o f the tank l i d and bottom w i l l be m a i n l y  changes i n the alignment o f the r e s o n a t o r Suppose the r e s o n a t o r w i t h  in Fig.8(b).  Taking  responsible  segments.  a d e f l e c t e d arm can be approximated as shown  a constant  t i p l o a d i n g c a p a c i t y , a new resonant  i s c a l c u l a t e d from eqn. (2.1), which i s , i n f a c t , a t r a n s c e n d e n t a l 1 j U ) C  z  TIP  with z  z  Z  ( n )  input  a  )  input  =  - 3Z o J  ( 1 )  equation  (n) input  (2.1)  „(n-l) , . „(n) i + JZo ' t a n -ccs_o input <n) (n-l) o input c  t \ -  (n) Z o  frequency  J  ;  +  j  z  t  a  n  tan^s c  £ s = — n n >> 1 A s m a l l computer program, c a l l e d DETUNE, w r i t t e n t o s o l v e t h i s e q u a t i o n the f o l l o w i n g r e s u l t s - F i g . 9. evacuation  I f the d e f l e c t i o n +Ag c o r r e s p o n d i n g  o f the vacuum tank i s known,  s h i f t due to changes i n b a r o m e t r i c  one can then e s t i m a t e  pressure.  t o the  the frequency  It i s fully justified  t o assume  t h a t the t i p l o a d i n g c a p a c i t y i s not i n f l u e n c e d by changes i n p r e s s u r e . rough e s t i m a t e  gave  A  has shown t h a t a 1.27 cm d e f l e c t i o n o f the h o t arm t i p r e s u l t s  i n an i n c r e a s e o f the a c c e l e r a t i n g gap by 5.2 x 10  5  m.  Then the r e s u l t a n t  frequency s h i f t due t o b o t h the change i n the t i p l o a d i n g c a p a c i t y and the resonator  l e n g t h i s n e g l i g i b l e compared to t h a t caused by the change i n  - 19 characteristic  2.2  Detuning  impedance.  of Resonator Due  to Temperature V a r i a t i o n  A change i n temperature causes the r e s o n a t o r panels contract, respectively. vacuum tank,  S i n c e the r o o t p i e c e s are f i r m l y a t t a c h e d to the  the expansion  (or c o n t r a c t i o n ) of r e s o n a t o r segments can  o n l y i n the d i r e c t i o n of the a c c e l e r a t i n g gap. d e f l e c t i o n of the hot arm resonant  frequency  to expand or to  i s s m a l l and  I t i s supposed t h a t the  can t h e r e f o r e be n e g l e c t e d .  o f the r e s o n a t o r o s c i l l a t i n g  occur  The  i n a p u s h - p u l l mode i s g i v e n  by  —*—  tan ^1=  Z ci  c  o^ where Z  o  resonant  (2.2)  (si C. T I P  i s the c h a r a c t e r i s t i c impedance of r e s o n a t o r segment, f frequency,  % i s the hot arm  (2.2)  Assuming co =  CO(Z  ,C^^.p, £) , we  q  the f o l l o w i n g e x p r e s s i o n  A f ^  A Z  Q  Z  Q  +  and w i s the  o b t a i n from the f i r s t  (£ - c/4f ) Q  A C  T  I  P  T  I  P  fo  +  i  TT  Al  „  C  I s i n 2g»*  +  *  s i n 2-°-Ji c  I t i s easy  to show t h a t AC.  TIP  C  TIP "  ^2. Z  „  D  s  Aft d  =  2fr  i s the  l e n g t h , C^j-p i s the t i p l o a d i n g c a p a c i t y  per r e s o n a t o r segment, 2d i s the a c c e l e r a t i n g gap width.  o  Aw W  _Aw w  S u b s t i t u t i n g i n t o the above e x p r e s s i o n we  a r r i v e at  resonator  d e r i v a t i v e of  eqn.  - 20 -  Af  A£  K  where K =  .1 s i n 2 — % c_  +  £ sin 2 ^  +  =  c  So as a f i r s t  Trd TTJL  A  o r d e r approximation we  f  d_  get  (2.3)  £  T h i s l e a d s t o the c o n c l u s i o n t h a t the frequency s h i f t s  caused by s m a l l changes  i n the r e s o n a t o r w i d t h are compensated by the frequency s h i f t s due simultaneous  changes i n the t i p l o a d i n g c a p a c i t y .  r e s o n a t o r i s r a i s e d u n i f o r m l y by an amount AT, i n c r e a s e r e l a t i v e l y by an amount aAT, expansion.  The  to  I f the temperature  the hot arm  length w i l l  a b e i n g the c o e f f i c i e n t  expected frequency i s then determined  o f the  of  linear  from  Af -  For  aAT  aluminum, a = 2.55  temperature  (2.4)  x 10 / ° C . 5  The maximum frequency s h i f t f o r a g i v e n  change can be found i n T a b l e I I I .  TABLE I I I Percentage r e s o n a t o r frequency change vs temperature changes of r e s o n a t o r p a n e l s  Af/f  (%)  AT  -1.3  x  10~  3  0.5  -2.6  x  10"  3  1.0  -3.9  x  10"  3  1.5  -5.2  x  10~  3  2.0  -6.5  x  10~  3  2.5  (°C)  - 21 C a l c u l a t i o n shows t h a t eqn. harmonics If  of the  (2.4) i s v a l i d  and  third  RF.  the d e f l e c t i o n o f the hot arm,  as the temperature  be n e g l e c t e d , the o v e r a l l frequency s h i f t s h i f t s due  f o r both the f i r s t  i s raised,  i s g i v e n by a sum  to the hot arm expansion and due  cannot  of frequency  to the d e f l e c t i o n .  For a s m a l l  d e f l e c t i o n the change i n the hot arm l e n g t h caused by a d e f l e c t e d arm i s s m a l l compared w i t h the change i n l e n g t h caused by an expansion. t i o n any change i n the l e n g t h can be n e g l e c t e d because  For a l a r g e  the frequency  deflec-  shift  caused by the d e f l e c t i o n i s at l e a s t an o r d e r of magnitude g r e a t e r than t h a t due  to the change i n l e n g t h .  To e s t i m a t e the frequency changecaused  d e f l e c t i o n the p a r a b o l i c a p p r o x i m a t i o n o f the d e f l e c t e d arm  by  the  i s considered.  The r e s u l t s were o b t a i n e d by means of the program DETUNE where the p a r a b o l a was  first  approximated  technique was  2.3  used.  by a s e r i e s of s t r a i g h t l i n e s and then the same  The graph i n F i g . 9 summarizes the  V o l t a g e V a r i a t i o n Due In  results.  to Temperature V a r i a t i o n  a d d i t i o n to frequency d e t u n i n g e f f e c t caused by r e s o n a t o r expansion  temperature w i l l produce  v a r i a t i o n s w i l l a l s o cause changes i n r e s o n a t o r r e s i s t i v i t y . changes i n r e s o n a t o r peak v o l t a g e .  the r e s i s t i v i t y  i s dependent on temperature  p = P  In the f i r s t  approximation  as  (1 + 3AT)  q  (2.5)  where P i s the s p e c i f i c r e s i s t i v i t y  and  resistance.  of the r e s o n a t o r r i s e u n i f o r m l y by  Should the temperature  3 i s the temperature  coefficient  amount AT, the d i f f e r e n c e i n the r e s i s t i v i t y would amount to  Ap -  The  simplified  p  o  This  BAT  e x p r e s s i o n f o r a r e s o n a t o r power l o s s i s g i v e n by  an  of  - 22 -  P  =  2w<5 a  where V  Q  -  i s the r e s o n a t o r v o l t a g e peak, w^ i s t h e average w i d t h o f  r e s o n a t o r segment and <5 i s the s k i n depth.  P  =  T a k i n g 3 = 4 x 10 Dee), V  Q  = 10  5  P  -3  (1  o  T h i s e q u a t i o n can be r e w r i t t e n as  + g AT)  for silver, P  (2.6)  Q  = 4.86 x 10  5  W f o r the main c y c l o t r o n (one  V and assuming a c o n s t a n t power i n p u t e q u a l t o P  q  i t is  p o s s i b l e t o e s t i m a t e t h e change i n r e s o n a t o r v o l t a g e caused by a temperature i n c r e a s e AT. from eqn. (2.7) V  V =  (2.7) /l +  3AT  The r e s u l t s a r e p r e s e n t e d i n T a b l e IV. TABLE IV Resonator v o l t a g e v a r i a t i o n caused by temperature v a r i a t i o n AT  2.4  (°C)  AV (V)  AV/V  0.5  -100  -0.1  1.0  -200  -0.2  1.5  -299  -0.3  2.0  -398  -0.4  2.5  -496  -0.5  (%)  Frequency D e t u n i n g Due t o E l e c t r o s t a t i c and M a g n e t i c F o r c e s The r e s o n a t o r segment can be approximated by a number o f p a r a l l e l p l a t e  c a p a c i t o r s w i t h d i f f e r e n t amounts o f charge r e s i d i n g on t h e s u r f a c e s of each  capacitor.  The e l e c t r o s t a t i c energy of such a system may e a s i l y be e v a l u a t e d  - 23 as soon as one knows the charge d i s t r i b u t i o n i n the r e s o n a t o r . when t h e number o f c a p a c i t o r s goes t o i n f i n i t y , integral. eqn.  In the l i m i t  one can r e p l a c e the sum by an  The e l e c t r o s t a t i c energy i s then c a l c u l a t e d a c c o r d i n g t o  (1.17).  Since i t holds  Z  =  that  cC g  where g i s the r e s o n a t o r gap and e i s the d i e l e c t r i c c o n s t a n t , and because the s u r f a c e s o f the r e s o n a t o r a r e maintained e x t e r n a l sources  a t f i x e d p o t e n t i a l s by means o f  o f energy, the t o t a l f o r c e  9  a c t i n g between two r e s o n a t o r  panels i s dW^ F  The  =  E  w  1 +  * 7  dg  2^1  sin  1 e s  2^ c  (2.8)  'oi  average e l e c t r o s t a t i c f o r c e on the r e s o n a t o r p a n e l i s 2TT/OJ F  E  V  - 2 7 {o E F  S  ^  ±  d  t  =  \  F  (2.9)  E  In the case o f RF f l a t - t o p o p e r a t i o n , the e l e c t r o s t a t i c f o r c e i s g i v e n by s i n 2—4, sin 2—I c c (1 + n) I + — ^ — + OJto, 2-±-  1 e a w  F  E=  "  4  V  where n |^ 3/ i|» ^ i » =  v  0  0  0  V Q  3  a  r  e  t  ^  e  V  (2.10)  oi  fundamental and t h i r d v o l t a g e  amplitudes  i n the l i n e which i s a q u a r t e r wavelength l o n g , ojj i s the fundamental frequency  and 003 i s the t h i r d harmonic frequency.  However, the e l e c t r o s t a t i c  f o r c e a c t i n g on d i f f e r e n t u n i t areas of r e s o n a t o r s u r f a c e s i s not c o n s t a n t . I t depends on the d i s t a n c e from t h e r o o t . Besides  the e l e c t r o s t a t i c f o r c e , t h e r e i s a l s o a magnetic f o r c e a c t i n g  on the r e s o n a t o r p a n e l s .  T h i s f o r c e i s c a l c u l a t e d i n a s i m i l a r f a s h i o n as  the e l e c t r o s t a t i c f o r c e .  The t o t a l magnetic f o r c e a c t i n g on the r e s o n a t o r  - 24 p a n e l i s e q u a l to the e l e c t r o s t a t i c  -  f o r c e because o f the e q u a l i t y of magnetic  and e l e c t r i c e n e r g i e s at resonance.  At resonance at any p o i n t i n s i d e  the  r e s o n a t o r the t o t a l ' energy d e n s i t y i s constant  C ' | v ( x ) | + L ' | l ( x ) | = const. 2  (2.11)  2  where x i s the d i s t a n c e from the r o o t .  For t h i s reason the sum  of the e l e c t r o -  s t a t i c and magnetic f o r c e s a c t i n g on a u n i t a r e a of the r e s o n a t o r s u r f a c e i s c o n s t a n t and does not depend on the d i s t a n c e from the r o o t . between two  total force  r e s o n a t o r panels i s  F = F  The  The  E  + F  = 2F  H  E  t o t a l average f o r c e i s  F  " =1  E v a l u a t i o n of F  = E  F  F  for w  =  F  H  = 32 i n . , g = 4.1  i n . , £ = 121.7  i n . and  ct  V , = V ol  = 100 kV r e s u l t s i n F  1  3 V  = - 5.46  average f o r c e per u n i t area i s constant  Nt = - 1.23  lb.  S i n c e the  total  the d e f l e c t i o n of the r e s o n a t o r  panel  cW  due  to t h i s f o r c e F  can be c a l c u l a t e d as i f the d e f l e c t i o n were due  weight of the p a n e l . Ag where K = 3 x 10  -3  The d e f l e c t i o n of the p a n e l t i p K F  x io  is AF  3 V  hot arm  cm.  in./lb.  = 0.99  The x 10~  1 0  (2.12)  I t f o l l o w s then Ag  max  = - 3.7  change i n the t o t a l average f o r c e due 2  by  a V  -3 -9.4  i s given  Nt = 2.23  l e a d s to a frequency  to the  x 10~  3  lb.  x 10  -3  in. =  to t h i s  The d e f l e c t i o n of the  s h i f t Af = - 0.01  % (see F i g . 9 ) .  displacement resonator  Frequency  changes were computed u s i n g the computer program DETUNE (see S e c t i o n I I . 2 . 1 ) .  - 25 2.6  Tolerances  t o be Met  To m a i n t a i n the r e s o n a t o r tuned resonant  frequency,  controlled. reference.  f o r the d e s i r e d o p e r a t i n g c o n d i t i o n s , the  v o l t a g e amplitude  The sensors w i l l  and v o l t a g e phase i n both Dees must be  r e f e r e n c e these q u a n t i t i e s t o some e x t e r n a l  The RF system w i l l be kept  i n tune by means o f s e v e r a l  servosystems.  T h e i r f u n c t i o n i s to h o l d the proper phase r e l a t i o n between the Dee v o l t a g e s , to h o l d a c o n s t a n t r e s o n a t o r frequency at  and t o keep t h e f i n a l  the maximum e f f i c i e n c y a t the r e s o n a t o r f r e q u e n c y .  operation i t w i l l  a l s o be n e c e s s a r y  1 1  amplifier operating  During  the RF f l a t - t o p  to c o n t r o l the phase r e l a t i o n and frequency  r a t i o o f b o t h harmonics. The of  RF v o l t a g e w i l l be determined  p a r t i c l e s u s i n g beam probes.  p r e c i s e l y by measuring the energy g a i n  C a p a c i t i v e p i c k - u p probes w i l l  a l s o be  i n s t a l l e d and they w i l l p r o v i d e us w i t h a c e r t a i n i n f o r m a t i o n on the l e v e l o f the RF v o l t a g e .  The a l t e r n a t i v e method o f measuring the RF v o l t a g e would be to  measure the Coulomb f o r c e e x e r t e d on a g i v e n s u r f a c e S by an a l t e r n a t i n g Vsinbot.  T h i s f o r c e i s g i v e n by  F  The  field  "  4  d e f l e c t i o n would be converted  reference point.  i n t o a s i g n a l t o enable  However, the p o s s i b i l i t y o f u s i n g b o t h  comparison w i t h some c a p a c i t i v e probes and  probes measuring the d e f l e c t i o n due t o the Coulomb f o r c e f o r a p r e c i s e d e t e r m i n a t i o n o f the RF v o l t a g e i s h i n d e r e d by two f a c t s .  First,  an a c c u r a t e  and permanent c a l i b r a t i o n i s d i f f i c u l t t o a c h i e v e because o f a p o s s i b l e motion of  the ground arm and h o t arm t i p s .  The computer r e s u l t s i n d i c a t e t h a t f o r a  c o n s t a n t r o o t c u r r e n t the t i p v o l t a g e changes by ± 3 % when a ±6 mm t i p d e f l e c t i o n occurs. assumed.  A l i n e a r approximation  o f the d e f l e c t e d h o t arm i s  Second, the c a l i b r a t i o n i s i n f l u e n c e d by the asymmetry i n the  p o s i t i o n o f a v i r t u a l grounding A frequency  p l a n e between the Dees.  s y n t h e s i z e r w i l l be used f o r e x c i t a t i o n o f the fundamental  - 26 and  the t r i p l e r w i l l d e r i v e the t h i r d harmonic f r e q u e n c y .  resonant frequency w i l l be h e l d constant b e l l o w s mounted at the r o o t . holding The  -  The  by  The  a d j u s t i n g the p o s i t i o n of  a c t u a l i n tune s t a t e w i l l be  a zero phase s h i f t between the r o o t c u r r e n t and  t h i r d harmonic v o l t a g e must be  fundamental  s h i f t e d by 180  attained  the loop  deg w i t h  tuning by  current.  respect  to  the  fundamental v o l t a g e . To a c h i e v e  the a c c e l e r a t i o n of i o n s w i t h o u t any  p a t h , the v o l t a g e as p o s s i b l e .  Simultaneously  lower r e s o n a t o r  by R i c h a r d s o n  the a c c e l e r a t i n g gap  the asymmetry i n the v o l t a g e s  must be  13  '  as  their  uniform  on the upper  segments should be minimized and kept c o n s t a n t .  12  allowable  d i s t r i b u t i o n along  adverse e f f e c t s on  and  Calculations le maximum  which are summarized i n T a b l e s  V, VI g i v e  the  tolerances. TABLE V Permissible  R  (in.)  l e f t - r i g h t voltage  asymmetry  15  25  35  50  60  0.5  0.5  0.9  1.9  2.7  i Su  (%)  where <5u  =  V, _ (R) - V . _(R) left right V . ( R ) + V . , fR) left right n  AV  The  = V  upper  (R) - V, (R) lower  d e s i r e d t o l e r a n c e s on the RF parameters are p r e s e n t e d  Section  1.3).  i n Table  I  (see  - 27 -  TABLE VI P e r m i s s i b l e top-bottom v o l t a g e  R (in.)  AV (%)  3.  asymmetry  20  30  40  50  70  0.5  0.5  0.5 '  1.0  2.5  VOLTAGE BREAKDOWN 3.1  Sparking Two major problems c o u l d be encountered i n a c h i e v i n g t h e r e s o n a t o r  voltage l e v e l necessary multipactoring.  f o r acceleration.  Both s p a r k i n g and m u l t i p a c t o r i n g might l e a d t o a v o l t a g e  breakdown i n the r e s o n a t o r . sparking  These a r e s p a r k i n g and  We s h a l l f i r s t  look a t the p o s s i b i l i t y o f  i n the r e s o n a t o r .  According  to K i l p a t r i c k ' s d e f i n i t i o n } ^  considered necessary  f o r sparking.  " c u r r e n t due to f i e l d  In a d d i t i o n , energetic p a r t i c l e s are  r e q u i r e d t o i n i t i a t e a cascade p r o c e s s which i n c r e a s e s the f i e l d c u r r e n t s t o the p o i n t o f s p a r k i n g " .  The l a t t e r i s a l s o the reason  s p a r k i n g i n the vacuum can occur  a t lower v o l t a g e s  necessary  i s considered  sparking.  i f only f i e l d Sparking  electrically  emission  represents  V/cm and l e s s , i o n s p r e s e n t  gradients exist  s u r f a c e by t h e thermal energy.  the gap between two energy  At low g r a d i e n t s , o f the o r d e r  i n the gap supply  a cascade process  complete d i s s i p a t i o n o f  t o take p l a c e a c e r t a i n maximum  must be s u p p l i e d to one o f the m e t a l s u r f a c e s . 10  why  responsible f o r i n i t i a t i o n of  a spontaneous, abrupt,  F o r a cascade p r o c e s s  emission  than those which would be  s t o r e d energy f o r a g i v e n v o l t a g e a c r o s s  metal s u r f a c e s .  emission i s  the energy, b u t when h i g h  i s due t o p a r t i c l e s r e l e a s e d from a m e t a l  - 28 K i l p a t r i c k presents no  sparks  is  excluded) :  a c r i t e r i o n which determines a t h r e s h o l d below which  should be observed i n vacuum (Presence  1 . 7 X 1 0  W I = W E e o 2  fields  5  « 1.8 x 10  E  o f e x t e r n a l magnetic  (3.1)  lh  W i s the maximum p o s s i b l e energy o f a p a r t i c l e a t the e l e c t r o d e s u r f a c e to a spark i n eV and E i s the f i e l d  g r a d i e n t i n V/cm.  To f i n d  prior  the maximum RF  energy W, a q u a n t i t y V* r e p r e s e n t i n g the h i g h e s t energy, n o n - r e l a t i v i s t i c m u l t i p a c t o r v o l t a g e f o r TT deg t r a n s i t 2  V* =  2gTT _ -  time i s g i v e n by  2 N.R.  |  (3.2)  1  where g i s the gap i n cm, X i s the wavelength i n a f r e e space i n cm, q i s the 2 charge o f a p a r t i c l e and m c Q  V/V*  i s the r e s t energy of a p a r t i c l e .  < 1 f o r a g i v e n RF v o l t a g e peak a c r o s s  evaluated  u s i n g the e x p r e s s i o n ZT7 2 W = 2 m c  considered  separately  the gap, the energy W can be  N.R.  2  Four d i f f e r e n t gaps a r e p r e s e n t  (3.3)  i n the TRIUMF r e s o n a t o r  (see T a b l e  and they a r e  VII).  I t i s not known i n advance what k i n d o f i o n s w i l l be p r e s e n t however, one can c o n s i d e r protons the lowest favourable.  f a c t o r V*.  I f the r a t i o  i n the r e s o n a t o r ,  which g i v e the extreme upper energy W and  I n o t h e r words, f o r h e a v i e r  S u b s t i t u t i n g the c a l c u l a t e d v a l u e s  i o n s the s i t u a t i o n i s more  o f E and W i n eqn.  r e s u l t s i n v a l u e s which a r e s m a l l e r than 1.8 x IO *. 11  In the case o f a 1 i n .  gap  the energy W c o u l d n o t be e v a l u a t e d because the r a t i o V/V* > 1.  way  o f f i n d i n g W would have t o be c o n s i d e r e d .  t h i s case i s r o u g h l y  (3.1)  Another  However, i t i s b e l i e v e d t h a t  e q u i v a l e n t t o the 2 i n . gap f o r which V/V* i s o n l y  s l i g h t l y l a r g e r than u n i t y . F o r V/V* - 1 eqn. (3.3) g i v e s o n l y  approximate  values.  The r e s u l t s are p r e s e n t e d i n T a b l e V I I .  to be expected  A c c o r d i n g l y , no sparks a r e  s i n c e the c o n s i d e r e d g r a d i e n t s a r e lower than those f o r which  a spark w i l l o c c u r as g i v e n by the K i l p a t r i c k ' s  criterion.  TABLE V I I K i l p a t r i c k ' s c r i t e r i o n a p p l i e d to TRIUMF r e s o n a t o r geometry  6  g =  2Q0  V  (kV)  E  (kV/cm)  X  (cm)  m c o  2  v*  3.2  200  g =  1  in.  100 39.4  1323.9  1323.9  1323.9  1323.9  939.3  939.3  939.3  939.3  1562.0  731.0  174.0  43.5  4.1  o  100  in.  39.4  0.13  0.14  1.15  1.02  0.54  * 9.16  (keV)  W I  2  g =  9.6  V/V* W  4.1 i n .  g =  13.1  (MeV) (kV)  in.  x  io  5  1.0  x  10  1.9  3  x  10  2.30 -  11  M u l t i p a c t o r i n g Ranges To i n v e s t i g a t e p o s s i b l e m u l t i p a c t o r i n g r e g i o n s i n the TRIUMF r e s o n a t o r 15  system a treatment pactoring  s i m i l a r to t h a t g i v e n by Smith  ( e l e c t r o n m u l t i p l i c a t i o n by secondary  f i e l d , t h e resonant  have a s i g n i f i c a n t  frequency,  Multi-  e m i s s i o n i n a vacuum) can  s t a r t as soon as s e v e r a l c o n d i t i o n s have been met. the e l e c t r i c  w i l l be a p p l i e d .  Many parameters such as  the r e s o n a t o r shape and o t h e r s  r o l e i n d e f i n i n g the m u l t i p a c t o r i n g r e g i o n .  I f we have two metal s u r f a c e s i n a vacuum the time o f f l i g h t o f an e l e c t r o n t ^ between the e l e c t r o d e s u r f a c e s must be an odd m u l t i p l e o f a h a l f p e r i o d o f the RF o s c i l l a t i o n  - 30 -  (2n  =  - 1) Tr  1 for  the g i v e n  co  frequency, voltage  and the gap f o r m u l t i p a c t o r i n g  second c o n d i t i o n i s t h a t the energy gained s u f f i c i e n t l y high For  coefficient  copper the energy range g i v i n g the secondary e m i s s i o n < 1500 eV.  q  the b u i l d - u p  be r e a c t i v e i n c h a r a c t e r effects.  respect  T  The l o a d p r e s e n t e d  using the f o l l o w i n g  0.0725 ( f )  and o f t e n p r e v e n t s  t o the g e n e r a t o r  L  where V ,  V  T  TU  f o r a copper  formulae^  2  1 2 D-^r}. l  <  i -  8  5  V  [(2n -  l  2  TL  (3.6)  denote the upper and lower t h r e s h o l d m u l t i p a c t o r i n g  multipactoring  order ^  voltages.  i s found from l/  n S  may a l s o  B  {1 +  V  greater  to the RF v o l t a g e .  The lower and upper t h r e s h o l d m u l t i p a c t o r i n g v o l t a g e s  ,  than u n i t y .  and t h e r e f o r e m u l t i p a c t o r i n g may l e a d t o detuning  m a t e r i a l can be e s t i m a t e d  The  coefficient  a heavy l o a d on the g e n e r a t o r  o f the Dee v o l t a g e .  greater  The energy g a i n o f the e l e c t r o n a l s o  depends on the i n i t i a l phase o f t h e e l e c t r o n w i t h Multipactoring u s u a l l y places  The  by the e l e c t r o n i n t r a n s i t must be  to g i v e the secondary e m i s s i o n  than one i s 200 eV < W  to s t a r t .  V  u V  2  TL  (3-7)  2  where V  q  D  , t h e energy p i c k e d  up by t h e e l e c t r o n c r o s s i n g t h e gap, i s g r e a t e r  than 100 eV f o r most m a t e r i a l s . In the TRIUMF r e s o n a t o r g = 6 in.  system the f o l l o w i n g gaps a r e p r e s e n t :  a c c e l e r a t i n g gap o u t s i d e  4 i n . nominal r e s o n a t o r 3 in.  CR  gap  h o t arm t i p t o grounding p l a n e  (one Dee  resonator)  - 31 -  The f i r s t  2 in.  a c c e l e r a t i n g gap i n CR  1 in.  hot arm t i p t o c e n t r e p o s t  (at i n j e c t i o n  and the t h i r d RF harmonics have been c o n s i d e r e d  point) separately.  TABLE V I I I ; T h r e s h o l d m u l t i p a c t o r i n g v o l t a g e s f o r the RF  Gap ( i n . )  V  TL  (  V  )  V  TU  (  V  fundamental  )  Order  6  720  1340  1  4  320  590  1  3  180  330  1  2  80  150  -  1  20  40  -  TABLE IX T h r e s h o l d m u l t i p a c t o r i n g v o l t a g e s f o r the t h i r d  Gap ( i n . )  6  V  TL  (V  >  v  T U  (V)  harmonic  Order  6500  12100  1  2510  12100  2  1530  12100  3  1090  12100  4  2880  5360  1  1110  5360  2  1620  3000  1  620  3000  2  2  740  1360  1  1  180  330  1  4  3  - 32 No h i g h e r o r d e r r t m l t i p a c t o r s can occur except o f r i s e of 1000  those i n d i c a t e d above.  V/usec i s u s u a l l y c o n s i d e r e d s u f f i c i e n t  r e s o n a t o r m u l t i p a c t o r i n g under good vacuum c o n d i t i o n s . i s a v a i l a b l e i t i s p o s s i b l e to break through  -k  through  I f t h i s r a t e of  rise  the m u l t i p a c t o r i n g range at poor  -5  p r e s s u r e s of the o r d e r 10 3.3  to pass  A rate  - 10  Torr.  Rate o f R i s e of Resonator V o l t a g e The  r a t e of r i s e of r e s o n a t o r v o l t a g e i s a s s o c i a t e d w i t h problems of  multipactoring.  Knowing the t r a n s i e n t b u i l d - u p of r e s o n a t o r v o l t a g e  a l l o w s us to p r e d i c t whether any d i f f i c u l t i e s w i l l be encountered s w i t c h on p r o c e s s .  An i n v e s t i g a t i o n was  amplitudes  during a  c a r r i e d out i n o r d e r to compare the  r a t e of v o l t a g e r i s e of r e s o n a t o r f e d d i r e c t l y by a g e n e r a t o r w i t h t h a t of a r e s o n a t o r f e d v i a a resonant  transmission l i n e .  For t h i s an  equivalence  between a r e s o n a t o r w i t h d i s t r i b u t e d parameters and a lumped parameter r e p r e s e n t a t i o n i s assumed. A r e s o n a t o r and  a generator  are coupled  as shown i n F i g . 1 0 ( a ) , where R  is  the i n t e r n a l r e s i s t a n c e of the g e n e r a t o r , V  is  the resonant  frequency  of the r e s o n a t o r .  lumped c o n s t a n t s as a s e r i e s resonant any  i s the g e n e r a t o r v o l t a g e and  The  circuit.  c o u p l i n g network between the g e n e r a t o r  and  resonator i s represented We  do not wish to i n t r o d u c e  the r e s o n a t o r as we  i n t e r e s t e d i n the r a t e o f r i s e of r e s o n a t o r v o l t a g e amplitude the r e s o n a t o r w i t h a c o u p l i n g network. equivalent  circuit  For t h i s reason we  r e s i s t a n c e and v o l t a g e o b t a i n e d by t r a n s f o r m i n g R the c o u p l i n g on the r e s o n a t o r frequency The  new  The  r e s o n a t o r resonant  generator  w i l l use  frequency  impedance R  E  CJ  G  , V  G  v i a M.  i s i n c l u d e d i n a new  i s co  °  =  are  without  E E shown i n F i g . 10(d),where R , V_ are e q u i v a l e n t CJ  by  The  loading  the generator effect  capacitance  of C.  /Ec  i s chosen so t h a t the r e s o n a t o r i s matched f o r  to  - 33 only f = f . Q  At t h i s p o i n t i t h o l d s t h a t R = R^.  steady s t a t e r e s o n a t o r amplitude E V V„ = 2 -v. i  at f = f  be V  s r  The c o n d i t i o n t h a t the 100 kV g i v e s  r  S  n  w  (3.8)  t  where Q i s the q u a l i t y f a c t o r of the unloaded frequency  o f the g e n e r a t o r .  According  r e s o n a t o r and to = 2frf i s " the  to K i r c h h o f f ' s law f o r our c i r c u i t  we  can w r i t e i f E Idt + R I + R I  AT L  dt  C  Assuming the system i s i n i t i a l l y Laplace transforms  V v = 2 -r-^ s i n tot r c  q u i e s c e n t , i . e . V„(0) = 0, 1(0)  (3.9)  0 and t a k i n g  results i n  V„(s)  (3.10)  P(s)  where P ( s ) = (to + s ) ( L C s 2  V  The  =2  v  2  2  + RCs + R^Cs + 1)  c  p o l e s of V (s) are  s, „ = ±ito 1.2 J  3,4  Having o b t a i n e d the p o l e s the i n v e r s e t r a n s f o r m takes the f o l l o w i n g form 1  V co P(s) 0  s  t  e  ds  T h i s i n t e g r a l i s u s u a l l y s o l v e d by c a l c u l a t i n g  (3.11)  the v a l u e o f the r e s i d u e w i t h  r e s p e c t to each p o l e and summing over a l l p o l e s .  V.  o k=l x - [ P ( s ) ] ds s=s,  , k s  fc  I t f o l l o w s then  (3.12)  - 34 To e v a l u a t e V ~ ( t ) a s m a l l computer program was  written.  The r e s u l t s  summarized i n F i g . 11, which shows the r i s e o f r e s o n a t o r v o l t a g e d u r i n g the t r a n s i e n t f o r f = f determined used  from the power l o s s , energy  f o r computation  L = 8.515  = 22.66 MHz.  x 10~  (CRM  - one Dee  amplitude  f o l l o w i n g lumped  parameters  s t o r e d , resonant frequency have been  r e s o n a t o r ) : C = 5.794 x 10  H, R = R^ = 1.709  8  The  are  x io" ft,, Q = 7091.  1 0  F  The energy  3  s t o r e d and  power l o s s f o r a s t e a d y s t a t e v o l t a g e amplitude of 100 kV have been computed by the program RESLINE.  S i n c e the q u a l i t y f a c t o r of a one Dee  resonator i n  the main c y c l o t r o n i s a l s o the same, the r a t e o f r i s e o f v o l t a g e  amplitude  would a l s o be g i v e n by the graph i n F i g . 11. To f i n d the t r a n s i e n t response o f the r e s o n a t o r v o l t a g e amplitude when the r e s o n a t o r i s f e d v i a a resonant r e p r e s e n t a t i o n of our system  t r a n s m i s s i o n l i n e the f o l l o w i n g e q u i v a l e n t  can be drawn - F i g . 10(b).  V  = V Gr  generator v o l t a g e , R  i s the i n t e r n a l r e s i s t a n c e of the g e n e r a t o r , co i s the  g e n e r a t o r frequency and U ) Rj,  Cj,  and R ,  C ,  2  L  2  i s the i n i t i a l  q  R  I  5  resonant frequency o f each  are lumped parameters  2  resonant l i n e , r e s p e c t i v e l y . resistance at f  V i  +  R  2  = G  < ' >  R  3  dlv M  Supposing  ^  I dt +  (L  2  2  di -  M  ^  the system  transforms gives  from t h i s  a n a l y s i s y i e l d s the f o l l o w i n g two  c7  +  the  i s matched t o the g e n e r a t o r i n t e r n a l r e s i s t a n c e , i . e .  4f  circuit  o f the r e s o n a t o r and  circuit.  I t i s assumed t h a t a steady s t a t e i n p u t  The mutual i n d u c t a n c e , M = k / l ^ L ^ , i s determined The  s i n ojt i s the O  + M ) ^ i  dl  +  ( L 2  +  M ) Z  JL  is initially  +  2  _  condition.  equations  - M | ^ + M— 2  1  13  =  0  (3.14)  f I dt + 2  (R  2  + R )I G  2  = V  G  q u i e s c e n t the s u b s t i t u t i o n of L a p l a c e  (3.15)  - 35 -  V  ( ) = —  v  (co + s ) ( A s  l  M C O C " C J | S 2  2  2  2  ° + A s  S  u  -  + A s  2  — + A^s + A ) ( 1  /  3  (3.16)  k)  -  5  z  where  A  -1  "iQ A„ = —  +  2  u 2  Ql —  R  +  G  Q^gCl - k )  2  L (l- k )  2  w w 2  2  2  + Q!Q (w + w )  RgUj  2  2  2  0.^(1  3  2  - k )  QjLgd  2  3 co (co Q + 0) Q ) 1  A, =  2  2  2  1  R co  1  +  2  QjQid 2  r  - k )  -  k ) 2  2  Y~ L (l - k ) 2  2  co  2  A. 1  5  -k  2  In o r d e r to c a l c u l a t e the i n v e r s e t r a n s f o r m of V  (s) we C  of V  (s). C  S i n c e o n l y two  must know the p o l e s  l  out o f s i x e x i s t i n g p o l e s c o u l d be  evaluated  l  n u m e r i c a l l y the computer was  used to f i n d the remaining  i n v e r s e t r a n s f o r m can be e v a l u a t e d 6  V,  (t) = I 1  four poles.  The  as  - V McoC ojfco s o 2 1 2 k 2  0  2  e  S k t  (  3  >  l  y  )  d7r < >W  k = 1  p  8  k where P ( s ) denotes the denominator of eqn. r e s o n a t o r v o l t a g e amplitude found  (3.16).  amplitude  at BB'  The  overall rise  d u r i n g the t r a n s i e n t f o r f = f^ = f  to be the same as t h a t p l o t t e d i n F i g . 11.  p l o t t e d i n F i g . 12.  The  amplitude  The  2  = f  of was  i n i t i a l voltage r i s e i s  o f the g e n e r a t o r v o l t a g e , V  , i s twice  the  (see F i g . 10(b)) f i x e d by the program RESLINE f o r a matched  system. The  parameters used i n computation have been f i x e d as f o l l o w s .  fixed internal resistance R RESLINE was  = 2200 Q,  For a  ( t r i o d e a m p l i f i e r ) a computer program  run i n o r d e r to match the i n p u t impedance at f  = 22.66 MHz.  From  - 36 the v a l u e s  o f energy s t o r e d , power d i s s i p a t e d , q u a l i t y f a c t o r computed by the  (3.13)  program and u s i n g eqn. V  G  =  2 x 16422  V, Q  k = 1.048 x i o  ,  - 3  7091,  =  a  L  the parameters M , C , k c o u l d be Q  5473,  =  2  = 1.935 x 1 0  2  obtained:  2  _3  = 1.345 x 10  M  8  H, C  =  2  2.549 x 10~ F, 11+  H .  S i m i l a r a n a l y s i s was done f o r a r e s o n a t o r  c o n s i s t i n g o f two Dees, where  the r e p r e s e n t a t i o n i n terms o f lumped parameters shown i n F i g . 10(c) was used. Choosing the c u r r e n t meshes as shown i n F i g . 10(c) r e s u l t s  i n the f o l l o w i n g  equations. dl  Vl  +  1  1  +—  I,dt  1  1  +  I dt 2  M  dl 3 _  1  !  i d T - c:  + L  d I  _ M  r  i  +  I dt 0  2  d  t  1  1  c,  3  2  < 2 L  +  (  L  3  +  M  _ _  )  + M  f  +  I dt 3  dl,  +  dl,.  4  1  + — I„dt 2  I. dt  +V3  I dt  _  2  • k->  1  (3.18)  0  I dt  C"  +  d  >dt  (3.19)  0  =  dl,  (3.20)  '  ! I.dt  +  +  (R  3  +  RG)Ik  = -  V  (3.21)  G  3•  R^ , C j ,  and R , 2  respectively.  C^,  a r e lumped parameters of t h e f i r s t  L , C , R 3  3  Dee-to-Dee c a p a c i t a n c e . r  a r e lumped c o n s t a n t s  3  /  2 3  3  initial  frequency resonant  1^(^ f  f  Q  .  i s not, i n general,  this  g  c i r c u i t s each h a v i n g  were then connected together  L (C 2  2  + 2C^)  the resonant  3  frequency  case, however, a p r o p e r r e s i s t i v e  (for a d e t a i l e d explanation,  L C 3  line.  C^ i s a  I t must be  were c a l c u l a t e d a t the r e s o n a t o r  The t h r e e resonant  frequency  + 20^)  of the resonant  M = kv L~L„ i s the mutual i n d u c t a n c e .  understood t h a t the l i n e parameters L , C resonant  and second Dee,  the same  and d r i v e n a t f . Q  o o f t h r e e coupled  circuits.  In  i n p u t impedance i s a t t a i n e d o n l y at f  see S e c t i o n s  II.9.3 and II.10.2)  - 37 An i n i t i a l l y quiescent system i s again assumed.  A long and tedious  algebraical derivation gave the following equations f o r V _ (t) and V _ (t) R(s, ) e ^  8  V  (t) =  I—  (3.22)  k=l ^ ( B ) ]  S  where R(s) =  -  C  C  2  3  M V  s w(L  + R)  3  q  l S  (3.23)  2  P(s) = (co + s ) [ ( A 1 0 A 1 - A A ) s 2  2  g  + (A A 12  + (A A 12  + ( A  2  11  1 Q  2  + A A  2  l  + A A  +-A A  n  A  6  + (A A + A A 11  7  I Q  JQ  2  -  + (A A  3  1 2  4  3  + A ^  + A A )s + A A ] n  5  1 2  2  1  A„ = (C C 2  A  3  A  A  =  -  2  L  2  2  C  2  H "  C  1  2  + C)(LRC  1  +  2  L  1  2  lV  8  =  L  1  (  H  C  C  1  1  3  1  +  = R (C C 1  A  9  =  A  10=  H  H  C  L  3  C  3  l  1 2  R  1  2  R  C  2  (  + R C ) + R ( C  2  2  + R L C )  2  +  1  = M C C  6  7  A  (  H  5  A  H  C  H  = C (R C  k  A  TT  H  v  2  C  V  + C ) 2  x  C  2  2  H  C  1  5  Q  +  - Cj)  + A^A^s  2  (3.24)  5  A = C L L (CC + C ) 1  \A )s +  --AgA^s"* +  3  + A A )s  3  1  C  2  >  +  S  ^2  ~  V  - 38 -  l l  A  A  -  1 2  3  C  (  V  +  = 1  s^, s^, ... ,  are poles of R(s ) e  8  V„  3  R  (t) = I  ( s ) . The e x p r e s s i o n f o r  (t) i s  ^  3  —  (3.25) i t P ( s ) ]  k-1  k where R(s) = - M C V c o s [ ( C  + C )(R C s +  2  3  Q  tt  P ( s ) = (u> + s ) [ ( A A  1  + A A  2  2  1  1  6  g  1 2  s  1  )s  6  + (A A 2  + (A_A„ + A A + A A + A A 3  +  6  6  2  =  C  2  C  A„ = L C 3  3 3  A, = k  A  + A A  8  + A  2 1  (  2  R  2  L  3  3  -  +  G  R  L  2  +  2  R  3  2  3  2  + C ) 2  3  2  2  V  )  G  A  A  1 3  1 1 +  5  + C C R (R„ + R )  (R_ + R )C + R C  = C ^ C ,  L  12  6  3  + 1) - C^]  + A^  + A  + A  10  A  y  13  ^  + A^A^)  )s + A A ]  2  2 2  3  1 5  M)  + L C  G  3  n  = 1  6  2 1  + A^Ag + A A  5 A  + A  y  7  11  8  2  + (A A  h  8  + A A  + AA  3  3  y  6  A, = C C,(L L  A  3  1  + (A A  5  2  7  (A,A + A A  5  1  2  2  8  (3.26)  + A^A^ + A ^ s  + A A 9  )s  4  14  + A ^ s  s  5  3  2  (3.27)  - 39 C  A  A.. = C (R_R  1  14  = CC L 14 1  3  G 2  + R R ) + L 2 3 2  2  9  \5  2  =  R  A, = CR C 10 114 2  A, = C C 11 14 A  = C (L L 12 3 2 3  A, , = C„( R L 13  3  2 3  M) 2  + R L + L R ) G 2 2 3  A computer program was  w r i t t e n to e v a l u a t e the i n v e r s e t r a n s f o r m s V C  and V  S  ( t ) . The r e s u l t s have i n d i c a t e d t h a t V C  same r i s e t i m e s , both 11 and 12. line  (CRM)  o v e r a l l and  initial.  ( t ) and V  C  r e s o n a t o r a t the end o f the  lowers the i n p u t r e s i s t a n c e by a f a c t o r of 2.  v o l t a g e g e n e r a t o r must now  be  (t)  ( t ) possess the 2 The r e s u l t s are p l o t t e d i n F i g s .  l  Note t h a t r e p l a c i n g one Dee by a two Dee  r e s o n a t o r s t e a d y s t a t e amplitude of 100 kV,  l  In o r d e r to a t t a i n a  the a v a i l a b l e power from a matched  doubled.  Comparing the graphs i n F i g . 11 we  deduce t h a t the o v e r a l l r i s e o f r e s o n a t o r  v o l t a g e amplitude i s almost the same f o r b o t h the r e s o n a t o r f e d d i r e c t l y by a g e n e r a t o r and the r e s o n a t o r f e d v i a a resonant l i n e .  T h i s i m p l i e s t h a t the time  c o n s t a n t of the r e s o n a t o r t r a n s m i s s i o n l i n e system i s almost the same as t h a t of  the r e s o n a t o r f e d d i r e c t l y by a g e n e r a t o r .  However, the i n i t i a l  rate of r i s e  of  the r e s o n a t o r v o l t a g e amplitude i s l e s s when the c o u p l i n g between the r e s -  o n a t o r and the g e n e r a t o r i s a c h i e v e d by means of a resonant l i n e  ( F i g . 12).  T h i s r a t e of v o l t a g e r i s e i s l e s s than the d e s i r e d v a l u e o f 1000  V/usec.  voltage  source  can not s u p p l y an a d d i t i o n a l amount of power i n excess o f t h a t  needed f o r a steady s t a t e amplitude o f 100 kV, be p a s s e d . Appendix  A).  I f the  the m u l t i p a c t o r i n g range may  The lumped parameters were found by the program RESLINE (see  not  - 40 4.  -  FREQUENCY TUNING  4.1  Root S h o r t i n g P l a n e M o t i o n The  resonant  frequency  v a r i e s a c c o r d i n g to eqn.  of a r e s o n a t o r o p e r a t i n g i n the p u s h - p u l l mode  (1.5) where Z , C^-p, q  which i n f l u e n c e t h i s frequency.  Tuning  &  a  r  e  the p o s s i b l e v a r i a b l e s  of the r e s o n a t o r by  changing i t s  p h y s i c a l l e n g t h i s s u i t a b l e b o t h f o r f i n e and  coarse frequency  V o l t a g e u n i f o r m i t y a l o n g the a c c e l e r a t i n g gap  i s u n a f f e c t e d by moving the r o o t  shorting plane. [see eqn.  (1.19)].  determining resonator The  The  q u a l i t y f a c t o r s v a r y w i t h a changing Mechanical  f a c t o r s i n choosing  resonant  frequency  problems such as good c o n t a c t s w i l l be  the  t h i s k i n d of t u n i n g f o r the main c y c l o t r o n  system. graphs i n F i g . 13 p r e s e n t the r e s u l t s o b t a i n e d by s o l v i n g the above  mentioned t r a n s c e n d e n t a l e q u a t i o n . of l e n g t h f o r f i x e d v a l u e s of  4.2  adjustments.  The  frequency was  c a l c u l a t e d as a f u n c t i o n  and Qp-j-p*  C a p a c i t i v e E f f e c t Near the A c c e l e r a t i n g Gap. The  frequency  t u n i n g can be v e r y simply accomplished  l o a d i n g c a p a c i t y a t the h i g h v o l t a g e end transcendental equation  (1.5) w i t h C ^ p  i n the graph i n F i g . 14. capacitance e x i s t .  of the r e s o n a t o r .  by v a r y i n g the t i p S o l v i n g the  as a v a r i a b l e gave the r e s u l t s  S e v e r a l p o s s i b l e ways of i n t r o d u c i n g an  In o r d e r not to a f f e c t  plotted  additional  the v o l t a g e u n i f o r m i t y t h i s  a d d i t i o n a l c a p a c i t a n c e must be spread u n i f o r m l y along the a c c e l e r a t i n g gap. The was  resonant  frequency  c a l c u l a t e d when the  t i p l o a d i n g c a p a c i t a n c e was  c a p a c i t i v e p l a t e s connected The  q  to the ground arm  i n c r e a s e d by moving the  t i p s as shown i n F i g . 1 5 ( a ) .  f o l l o w i n g parameters were used i n c a l c u l a t i o n : p l a t e l e n g t h = 4 m,  width Z  of a ten s e c t i o n r e s o n a t o r made up i n h a l f - s c a l e  = .112  = 43.2  m,  Q, Q  p l a t e t h i c k n e s s = 0.012  m,  = 5180,  = 7 pF.  Q  = 9020, C  T I p  r e s o n a t o r l e n g t h = 1.52  m,  plate  - 41 The  resultant  frequency i s the same r e g a r d l e s s  c o n s i d e r e d i n the c a l c u l a t i o n . assumed.  F o r s i m p l i c i t y one upper r e s o n a t o r segment i s  An a d d i t i o n a l c a p a c i t a n c e due to the p l a t e i n the i n i t i a l  d^ = 0.012 m i s C = 2.41 pF. hot  o f the number o f s e c t i o n s  arm t i p the t i p l o a d i n g  C = 0.413  By moving the c a p a c i t i v e p l a t e down towards the  capacity  i s increased  0.0508 - d  the f o l l o w i n g  Z °  2  = 0.022 m. A computer program was  equation  =  t a n —I  by  0.0508  where d v a r i e s between d^ = 0.012 m and d used t o s o l v e  —-rr—-  C  TIP  U  +  . C  )  A new resonant frequency f r e s u l t s f o r each new v a l u e o f C. frequency change i s d e f i n e d f  Af = 100  The percentage  as  " f  l  f  l  where f ^ r e f e r s to the p l a t e i n the i n i t i a l p o s i t i o n .  F i g . 16 shows the  computed frequency change as a f u n c t i o n o f the c a p a c i t i v e p l a t e Ag  4.3  position  = d - dj.  Ground Arm T i p D e f l e c t i o n The  capacity  i n v e s t i g a t i o n of tuning also included  the r e s o n a t o r by changing the t i p  the d e f l e c t i o n o f the ground arm t i p s .  produced by d e f l e c t i n g the ground arm t i p s .  c h a r a c t e r i s t i c impedance o f the r e s o n a t o r near the t i p w i l l  two  e f f e c t s combine to g i v e  the new resonant f r e q u e n c y .  a p p r o x i m a t i o n the t i p l o a d i n g deflections  capacity  t h i s statement i s v a l i d .  the program DETUNE.  loading  Two e f f e c t s a r e  Both t h e t i p l o a d i n g  the  by  position  c a p a c i t y and  change.  These  I n the f i r s t  i s taken t o be c o n s t a n t .  For s m a l l  The new r e s o n a n t frequency was computed  The d e f l e c t e d ground arm t i p was approximated by b o t h  - 42 a straight  l i n e and a p a r a b o l a .  Two s e t s of c a l c u l a t i o n s were done. t u n i n g o f the CRM r e s o n a t o r .  0 3  =69.27 MHz. AJL  one r e p r e s e n t e d  = 38.3 ft, C  q  = 14 pF, f  T I p  i s the l e n g t h o f t h e d e f l e c t e d t i p .  1  *  B  p l o t t e d i n F i g . 17.  the t u n i n g  (see S e c t i o n I I I . 2 . 2 ) .  was done w i t h the parameters: I = 1.55 m, Aft^ = 0.4064 m, C q  = 42.5 ft. I n t h i s case the d e f l e c t i o n was accomplished  s e c t i o n s o f the ground arms as shown i n F i g . 15(b).  = 23.1 MHz  The r e s u l t s a r e  The o t h e r s e t o f c a l c u l a t i o n s s i m u l a t e d  i n v e s t i g a t e d on a h a l f - s c a l e r e s o n a t o r  Z  the coarse  The r e s o n a t o r was r e p r e s e n t e d by the f o l l o w i n g  parameters: I = 3.09 m, AJ^ = 0.81 m, Z f  The f i r s t  T I  p  This =  calculation  1 pF>  by u s i n g hinged end  The curves i n F i g . 18(a)  r e p r e s e n t the computed v a l u e s .  4.4  Tuning  Bellows  a t the Root  I t i s a l s o p o s s i b l e t o a l t e r the resonant o f the r e s o n a t o r .  frequency by changing  the volume  F o r a s m a l l change i n the r e s o n a t o r volume o c c u r r i n g at the  boundary,perturbation  theory lends i t s e l f  f o r c a l c u l a t i o n o f a new  resonant  frequency. Suppose t h a t a s m a l l p o r t i o n o f t h e r o o t plunger moved e i t h e r forward o r backwards.  (see F i g . 19) can be  One segment i n f u l l - s c a l e i s assumed: 1.6  Z^ = 38.3 ft, l = 3.09 m,  = 14 pF. < W  H  f = f .: 1 + 2 — o where  H  inwards,  >  ~  <W>  <  W  F  A new resonant  frequency  i s g i v e n by  >  (4.1)  -  i s the magnetic energy i n the volume o b t a i n e d by moving the b e l l o w s W i s the e l e c t r i c energy i n the same volume, W i s the t o t a l E  s t o r e d i n the r e s o n a t o r .  energy  The f o l l o w i n g assumptions are made i n o r d e r t o  s i m p l i f y our c a l c u l a t i o n : i)  The e l e c t r i c energy d e n s i t y i n the v i c i n i t y o f the r o o t w  ii)  E  plunger  = 0.  The magnetic energy d e n s i t y i n the volume V " i s the same as t h a t a t  - 43 the r o o t . iii)  The motion o f the b e l l o w s from the i n i t i a l  p o s i t i o n i s such t h a t the  r e s o n a t o r volume i s d e c r e a s e d , iv) It  The l e n g t h o f the r e s o n a t o r i s taken t o be I then f o l l o w s  c/(4f ). Q  that  1 f = f  1 +  4f §  sin  D  Ail  +  2^A£) c  (4.2)  where § c h a r a c t e r i z e s the percentage p o r t i o n o f the r o o t p l u n g e r s u r f a c e which was moved and AJl i s the t r a v e l of the b e l l o w s . c = 3 x 10  8  Given t h e v a l u e s f = 2 3 . 1 o  MHz,  m and § = 0.171 we a r r i v e a t sin  Af 1 + 0.052585 LI  +  2—Ail c  - 1  2% T h i s e x p r e s s i o n y i e l d s the percentage f r e q u e n c y change as a f u n c t i o n o f the p o s i t i o n o f b e l l o w s b o t h f o r the f i r s t  and t h i r d harmonics.  v a l u e s a r e p r e s e n t e d i n T a b l e X and p l o t t e d i n F i g . 20.  TABLE X Percentage frequency change v s p o s i t i o n of tuning bellows AJl  (m)  Af /f 0  Q  (%)  - 0.002  0.0105  - 0.004  0.0210  - 0.006  0.0315  - 0.008  0.0421  - 0.010  0.0526  The c a l c u l a t e d  - 44 4.5  T h i r d Harmonic Tuning  Diaphragms  A s e c t i o n of the t r a n s m i s s i o n l i n e , a q u a r t e r wavelength l o n g , s h o r t c i r c u i t e d at one end and open c i r c u i t e d a t the o t h e r one fundamental frequency of  resonates  a t the  g i v e n by the l e n g t h of the l i n e and a t a l l odd  t h i s fundamental frequency.  In the c y c l o t r o n two  Dee  harmonics  r e s o n a t o r t h i s would  r e p r e s e n t the push-push mode which can not be used f o r a c c e l e r a t i o n of i o n s . For the f i x e d v a l u e s of Z  and  C  o  m T T 1  the - c o n d i t i o n f o r o s c i l l a t i o n s of  the  TIP  r e s o n a t o r both at the fundamental and  the t h i r d harmonic f r e q u e n c i e s ( p u s h - p u l l  modes) i s t a n ^£ = c  3  (4.3)  tan ^ c  where co i s the fundamental frequency  and £ i s the hot arm  t h i s e q u a t i o n f o r an unknown hot arm  l e n g t h £ shows t h a t no  for  the v a l u e s of £ between 0 and -7-  One  concludes  length.  Solving  solution  exists  t h a t the r e s o n a t o r can  not  4 be e x c i t e d both i n the f i r s t  and t h i r d harmonic p u s h - p u l l modes s i m u l t a n e o u s l y  u n l e s s one uses some a d d i t i o n a l t u n i n g element. Two in  p o s s i b l e ways of i n f l u e n c i n g the resonant  e i t h e r a t o t a l capacitance or inductance  frequency. for  frequency  results i n a different  S i n c e the aim i s to retune e i t h e r the f i r s t  the r e q u i r e d frequency  exist.  A change resonant  or the t h i r d harmonic  ratio  (4.4) a t u n i n g element a f f e c t i n g one harmonic more than the o t h e r and p l a c e d at some d i s t a n c e from the r o o t was  sought.  Movable diaphragms, connected  arms, seemed to be the b e s t s o l u t i o n Two  54).  d i f f e r e n t methods were a p p l i e d i n order to f i n d out the c o r r e c t  p o s i t i o n and resonant  (see F i g .  to the ground  e s t i m a t e the detuning  frequency  separately.  e f f e c t of diaphragms on e i t h e r harmonic  F i r s t p e r t u r b a t i o n theory  was  used to  - 45 c a l c u l a t e the frequency diaphragms. where now is  The  new  -  v a r i a t i o n f o r d i f f e r e n t p o s i t i o n s and dimensions of  resonant  frequency  i s determined  a c c o r d i n g to eqn.  W' i s the magnetic energy i n a volume V'occupied n.  (4.1)  by a diaphragm,  the e l e c t r i c energy i n a volume V a n d W i s the t o t a l energy s t o r e d i n the  resonator.  For e v a l u a t i o n of e n e r g i e s s t o r e d i n a volume V  diaphragm c r o s s - s e c t i o n i s assumed to be r e c t a n g u l a r .  the t u n i n g  S i n c e the  field  d i s t r i b u t i o n i s u n i f o r m along the v e r t i c a l a x i s , the time average e n e r g i e s i n a volume V* are found by i n t e g r a t i n g eqns. (1.15) and  <w^> =|civ l c o  <w^>  where E, = g'/g, t = %  2  - £^  2  = |civ/  5  u  u  2  2  v+  (1.16)  s i n 2-f-l - s i n  -  2  (4.5)  ^o  sin 2 f k  V -  -  2  - sin  2  2 ^ (4.6)  g i s the r e s o n a t o r gap, % ' i s the h e i g h t o f the diaphragm,  i s the t h i c k n e s s of the diaphragm and  (£^ + 1^)12  i s the d i s t a n c e  of the c e n t r e of the diaphragm from the r o o t .  The  was  are the wavelengths f o r the  taken t o be £ = A /4 = 3A-/4, where A.., A  first,  t h i r d harmonic, r e s p e c t i v e l y .  average width  The  of the r e s o n a t o r segment.  by eqns. ( 4 . 5 ) , (4.6)  and  (1.18) i n t o  sin f = f  1 +  2?  2-°-£ c  l e n g t h of the  diaphragm width  resonator  i s e q u a l to  the  S u b s t i t u t i o n o f the e x p r e s s i o n s  (4.1)  given  l e a d s to  s i n 2-f SL L  2  l  (4.7)  2^ ^c F i g . 21 shows the v a r i a t i o n of f ^ , f the r e s o n a t o r .  3  f o r a v a r y i n g diaphragm p o s i t i o n  Af r e p r e s e n t s the change i n frequency  the diaphragm v e r t i c a l l y  caused  inside  by i n s e r t i o n o f  i n the r e s o n a t o r a t a g i v e n d i s t a n c e from the r o o t .  From the r e s u l t s one would expect  t h a t the diaphragm p l a c e d e i t h e r at  - 46  -  A / 2 4 o r 5 X / 2 4 from the r o o t s h o u l d make i t p o s s i b l e to Qi  01  r e t u n e the c a v i t y f r e q u e n c i e s so t h a t the c o n d i t i o n (4.4) t h i s c o u l d occur o n l y i f the i n i t i a l  Af, - = 1-3 i s s m a l l e r than Af  |3f . - f ' ol  otherwise  frequency  simultaneously i s met.  However,  d i f f e r e n c e (measured)  I  (4.8)  0 3 '  the frequency  change caused by t u r n i n g the  diaphragm would not s u f f i c e .  C a l c u l a t i o n s were done w i t h the g i v e n parameters:  Z  g = 0.1016 m,  q  f  = 38.3  Q, C  = 10.75  T I p  = 22.66 MHz,  f  oi  pF,  = 67.98  %' = 0.06  m,  t = 0.005 m,  MHz.  0 3  16  P e r t u r b a t i o n theory volume i s a l t e r e d .  assumes a s m a l l p o r t i o n of the o r i g i n a l  However, i f the h e i g h t g ' of the g i v e n diaphragm i s l a r g e ,  i . e . of the same o r d e r of magnitude as the r e s o n a t o r gap, p e r t u r b a t i o n theory must be used. between the diaphragm and frequency  and d i s t r i b u t e d parameters we  by f i n d i n g new  L, C parameters of the  at the p u s h - p u l l frequency  q  resonator.  r e p r e s e n t the r e s o n a t o r as a lumped parameter  f  = 22.66 MHz sin  where Z  resonant  c i r c u i t s w i t h lumped parameters  A t o t a l c a p a c i t a n c e of an unperturbed  C-  from a c a p a c i t y  the hot arm might predominate. A r e s u l t a n t  i s then determined  circuit.  a d i f f e r e n t method of  Additional effects arising  Supposing an e q u i v a l e n c e between the resonant  resonant  resonator  2-% sin  1  2Z c o  resonator  oscillating  i s (see S e c t i o n II.9.1) to  -r%  "  2  (4.9)  i s the c h a r a c t e r i s t i c impedance , SL i s the r e s o n a t o r l e n g t h and V  the r e s o n a t o r peak v o l t a g e . capacitance C  H  = 0.37  L e t the diaphragm r e p r e s e n t a low  pF at i t s h o r i z o n t a l p o s i t i o n and  a low  is  frequency frequency  c a p a c i t a n c e i n c r e a s e ACy when moved to the v e r t i c a l p o s i t i o n at a g i v e n p o i n t i n the r e s o n a t o r  (see F i g . 54).  of the r e s o n a t o r i s g i v e n by  An o v e r a l l c o n t r i b u t i o n t o a t o t a l  capacitance  - 47 s i n .-r^x sin  ^4  s i n .-j^-x (AC  y  -  C ) 2  s i n > where x i s the p o s i t i o n of the diaphragm as measured from the r o o t . hot-to-ground  arm  C ,  c a p a c i t i e s of a s e c t i o n where the diaphragm i s to be  i n the absence of the diaphragm.  The v a l u e s C , u  n  a good a c c u r a c y .  However, the v a l u e o f AC^  are  placed,  (L , C„ can be c a l c u l a t e d w i t h 1 2  should r a t h e r be measured as i t i s  mainly  a stray capacitance  (the t h i c k n e s s of the diaphragm i s v e r y s m a l l ) .  Af  f  f r e q u e n c i e s are g i v e n by  Q  «  the new  resonant  For  (4.10)  f„ = 2TT/L(C  H  +  c^)  1 V  2rr/L(C + Cf. + V  (4.11)  O rl  I n d i c e s H and V r e f e r to the h o r i z o n t a l , v e r t i c a l p o s i t i o n o f the diaphragm, respectively.  The  resonant  f r e q u e n c i e s f o r the t h i r d harmonic are  u s i n g the same formulae w i t h d i f f e r e n t parameters. we re used i n the c a l c u l a t i o n : A c ^ £ = 3.19  m,  g  = 0.0635 m,  shows a p e r c e n t a g e  change  = 6 pF  (measured),  t = 0.00635 m, i n resonant  w  = 0.635 m.  f o l l o w i n g parameters = 144 The  pF,C  diaphragm has  such f r e q u e n c i e s f ^ , f  Af  1-3  = 3f  First  ol  - f  o3  the l o a d e d r e s o n a t o r without  > 0  pF,  position.  geometry of the r e s o n a t o r as w e l l as by the c o u p l i n g network, two considered.  = 143.9  graph i n F i g . 22  f r e q u e n c i e s of the r e s o n a t o r are determined  cases must be  3  frequency when the t u n i n g diaphragm  has been moved from the h o r i z o n t a l to the v e r t i c a l S i n c e the resonant  The  obtained  by  the  different  a tuning  t h a t an i n e q u a l i t y h o l d s  (4.12)  - 48 -  I t f o l l o w s t h a t i f a t u n i n g diaphragm i s to be used, the f i r s t harmonic must be  a f f e c t e d much more than the t h i r d harmonic i n o r d e r  Assuming Af  t o s a t i s f y eqn. (4.4).  ^ 0.3 MHz an element i n t r o d u c i n g approximately  extreme p o s i t i o n should be used.  AC^ = 6 pF a t i t s  I t seems t h a t the b e s t p o s i t i o n f o r p l a c i n g  a t u n i n g diaphragm i n the r e s o n a t o r  i s at 1.9 m from the r o o t .  This p o s i t i o n  i s good f o r f = 22.66 MHz up t o f = 22.66 MHz + 3% b u t n o t f o r ol. ol f  Q  = 22.66 MHz - 3%.  I n t h i s p o s i t i o n the RF fundamental v o l t a g e amplitude i s  " o n l y " about 78 kV compared w i t h  96.5 kV a t 2.75 m from the r o o t  (see S e c t i o n  III.2.6). I f the resonant  Af  one  concludes  1-3  frequencies  = 3f  ol  - f  0 3  o f the loaded  < 0  t h a t the t h i r d harmonic frequency  resonator  s a t i s f y an i n e q u a l i t y  (4.13)  r e t u n i n g by means o f a t u n i n g  diaphragm i s d e s i r e d .  The b e s t p o s i t i o n i s now a t about 0.5 - 0.6 m from the  root.  |Af  Taking  the same  range o f f r e q u e n c i e s  f  | $ 0.3 MHz the p o s i t i o n would be good f o r the  = 22.66 MHz ± 3%.  - 49 5.  RESONATOR MODIFICATIONS 5.1  Extreme End Segments The o r i g i n a l l y proposed c i r c u l a r shape o f the vacuum  chamber demanded t h a t  the extreme end segments o f e i t h e r Dee be tapered t o f i t the vacuum R e s u l t s from p r e l i m i n a r y measurements  ( see S e c t i o n III.4.1)  tank.  indicated  t h a t the  tapered segments i n f l u e n c e d the e l e c t r i c a l p r o p e r t i e s of t h e whole r e s o n a t o r very s i g n i f i c a n t l y .  The q u a l i t y f a c t o r s were b a d l y degraded.  The reason f o r  the drop i n the q u a l i t y f a c t o r s seemed t o be due t o the t r a n s v e r s e c u r r e n t s i n the tapered segment caused  by a nonuniform c h a r a c t e r i s t i c impedance.  An  attempt was made to d e s i g n the tapered segment w i t h dimensions such  t h a t the  c h a r a c t e r i s t i c impedance would be c o n s t a n t  Suppose  throughout  the segment.  the dimensions o f the extreme end segment h o t arm are g i v e n as shown i n F i g . 23(a) ( a l l dimensions are g i v e n i n cm). found  The r e q u i r e d r e s o n a t o r gap i s  as f o l l o w s .  A sample  calculation: 20 3  b, = 20 + ^ ^ 1 1 = 22.5 cm 1  oo  The o r i g i n a l c h a r a c t e r i s t i c impedance c o r r e s p o n d i n g  In o r d e r t o lower  t o t h i s width i s  Z  t o o b t a i n Z = Z = 49 0, i t i s n e c e s s a r y 1 1 o r e s o n a t o r gap a t t h i s p o i n t i n the r e s o n a t o r t o equal  t o choose the  J  g  l  =  49  5.2 = 2.85 cm  The T a b l e XI g i v e s the r e q u i r e d dimensions t h a t r e s u l t i n a constant c h a r a c t e r i s t i c impedance.  T h i s change i n dimensions,  however, i n c r e a s e s t h e  power l o s s i n an extreme end segment compared to the l o s s o f a normal segment. The approximate c a l c u l a t i o n f o l l o w s , where t h e parameters c h a r a c t e r i z i n g a  - 50 normal segment are taken to be: V - V • o l Mhos/m, $ = 9.8 x 1 0 ~ eqn.  m,  6  f  o  = 100 kV, Z  o  = 49 ft, a = 5.78 x  = 46 MHz, >t = 1.55 m, w = 0.403 m. ' a.  10  7  A c c o r d i n g to  (1.12) the nominal power l o s s i n the r e s o n a t o r segment i s P = 14  kW.  C a l c u l a t e d v a l u e s f o r i n d i v i d u a l p a r t s of the m o d i f i e d extreme end segment are presented  i n T a b l e XI, where an average w i d t h , w = b., was ° a I  i n c r e a s e i n the power l o s s o f such a segment was found  taken.  The  to be AP = 6.5 kW,  with  the c u r r e n t d e n s i t y a t the r o o t going up by a f a c t o r o f 2. TABLE XI C a l c u l a t e d gaps and power  5.2  l o s s f o r a m o d i f i e d extreme end segment  i  b  1  22.5  89  2.85  7.00  2  27.6  71  3.55  5.23  3  32.7  60  4.15  3.64  4  37.8  52  4.80  2.46  5  40.3  49  5.20  2.10  (cm)  Z. (ft) l  g  ±  (cm)  P  ±  (kW)  C e n t r a l Region Segments The r e s o n a t o r segments i n t h e c e n t r e of the main c y c l o t r o n must be  m o d i f i e d to a l l o w the i n s t a l l a t i o n o f the c e n t r e column which w i l l support the main magnet and house an e l e c t r o s t a t i c i n f l e c t o r . must be obeyed when m o d i f y i n g i)  The b a s i c c r i t e r i a which  the c e n t r a l segments a r e :  A c e r t a i n minimum gap has to be p r e s e r v e d  to a v o i d any p o s s i b i l i t y of  sparking. ii) iii)  The q u a l i t y f a c t o r s s h o u l d be a f f e c t e d as l i t t l e  as p o s s i b l e ,  The v o l t a g e v a r i a t i o n along the a c c e l e r a t i n g gap and around the  - 51  -  c e n t r e post must s t a y w i t h i n the n e c e s s a r y The  last  two  requirements  S i n c e the contour  are aimed m a i n l y  t o l e r a n c e s (see Table I ) .  at the RF t h i r d harmonic o p e r a t i o n .  around the c e n t r e post i s f i x e d a c c o r d i n g to the  of beam dynamics, the p o s s i b l e m o d i f i c a t i o n s are l i m i t e d tips  o f the r e s o n a t o r p a n e l s .  l o a d i n g c a p a c i t y and  i)  the change i n the r e s o n a t o r  The  = 3f  and  1  The  the top view of r e s o n a t o r be as shown i n F i g . 2 4 ( a ) .  m and  = 1.433  m.  To  same resonance the r e q u i r e d C ^ p  C  C  A C  = 7.98  TIP  2  TIP  3  TIP *3  gap.  r e s o n a t o r i n h a l f - s c a l e i s assumed.  f o l l o w i n g parameters are g i v e n : Z  &j = 1.56  = 8.17 = 0.19 = 16  Q  = 40 ft, f  1  = 45.32  MHz,  tune these p a r t s of a-segment to the per segment width  pF  °TIP  pF  C  pF  A C  deg  TIP  i s g i v e n by eqn. = 18.8  pF  = 21.4  pF  =  pF  3  TIP  2.6  = 37  *3  3  deg The d i f f e r e n c e  t i p l o a d i n g c a p a c i t y f o r e i t h e r harmonic i n c r e a s e s w i t h  an i n c r e a s i n g f o r e s h o r t e n i n g angle tb .  S i n c e tj> i s v e r y l a r g e , s m a l l 3  changes i n the t i p l o a d i n g c a p a c i t y produce l a r g e changes i n f satisfy  the c o n d i t i o n of resonance.  c a p a c i t i e s can never be met find  ii)  (1.5)  1  <|> i s the f o r e s h o r t e n i n g angle f o r the t h i r d harmonic. i n the n e c e s s a r y  the  done to a s c e r t a i n a l l e f f e c t s a s s o c i a t e d w i t h  the r e s o n a t o r t i p s .  Let f  to the shaping of  T h i s i n c l u d e s the change i n the a c t u a l t i p  A d e t a i l e d a n a l y s i s was shaping  requirements  The  requirement  to  on t i p l o a d i n g  s i m u l t a n e o u s l y f o r b o t h harmonics.  some compromise i n ^ t h e requirements  3  f o r the t i p l o a d i n g .  One  could  Otherwise,  the need f o r an a d d i t i o n a l t u n i n g element a f f e c t i n g o n l y one  harmonic  becomes more i m p e r a t i v e i n o r d e r to o b t a i n a proper frequency  ratio,  We  want to f i n d  the e f f e c t on the resonant  frequency  or d e c r e a s i n g the r e s o n a t o r gap near the arm  tips.  of e i t h e r i n c r e a s i n g We  assume t h a t  C^p  - 52 Z ,  £  q  2  are constant.  The r e s u l t s o b t a i n e d w i t h the computer code DETUNE  show t h a t an i n c r e a s e i n the r e s o n a t o r gap near t h e c e n t r e post i n a p o s i t i v e Af due  and i n a n e g a t i v e Af .. The n e g a t i v e v a l u e o f Af~ i s  t o a l a r g e f o r e s h o r t e n i n g angle CJJ^.  were c l o s e t o ^ M  results  I f the l e n g t h o f the r e s o n a t o r  the same d e f l e c t i o n would cause the frequency  f ^ to  i n c r e a s e (see S e c t i o n I I . 2 . 1 ) . iii)  A d e f l e c t i o n o f e i t h e r the h o t arm o r ground arm t i p s i n f l u e n c e s the t i p l o a d i n g c a p a c i t y , as w e l l . t i p s r e s u l t s i n lower causes the frequency resonance. frequency  An i n c r e a s e i n the r e s o n a t o r gap near the  tip-to-ground capacity.  But a lower  f ^ to i n c r e a s e i n o r d e r t o s a t i s f y the c o n d i t i o n o f  As mentioned above s m a l l changes i n s h i f t s Afg.  produce l a r g e  However, as s t a t e d i n i i ) an i n c r e a s e i n the  r e s o n a t o r gap produces n e g a t i v e s h i f t s A f g . The r e s u l t a n t w i l l be g i v e n by a combination iv)  By d e f l e c t i n g  t i p capacity  frequency  of these two e f f e c t s .  the r e s o n a t o r panels we n o t o n l y s h i f t  the frequency b u t  we s i m u l t a n e o u s l y change the v o l t a g e d i s t r i b u t i o n near the c e n t r e p o s t . For g i v e n v a l u e s o f f , Z , l a p o s i t i v e d e f l e c t i o n o o 2.  (an i n c r e a s e i n the  r e s o n a t o r gap) g i v e s a h i g h e r v o l t a g e a t the h o t arm t i p , w h i l e the n e g a t i v e d e f l e c t i o n l e a d s t o a lower v o l t a g e . it  i s n o t p o s s i b l e to p r e d i c t  the f i n a l frequency.  of  changes  the v o l t a g e d i s t r i b u t i o n u n l e s s one knows  T h i s d i s t r i b u t i o n a l s o depends on whether the  m o d i f i e d segment i s connected frequency  I f the frequency  to the o t h e r segments o r not.  Both the  and v o l t a g e e q u i p o t e n t i a l s can be i n f l u e n c e d by the presence  o t h e r segments.  We conclude  t h a t i t i s v e r y d i f f i c u l t to p r e - c a l c u l a t e the r e q u i r e d t i p  l o a d i n g c a p a c i t y when e i t h e r the ground o r h o t arm t i p s a r e s i m u l t a n e o u s l y deflected.  The s o l u t i o n to t h i s problem would p r o b a b l y . y i e l d some optimum  v a l u e s o f the Q's and v o l t a g e u n i f o r m i t y b u t t h i s would n o t be the i d e a l s t a t e  - 53 when a l l segments are tuned to resonance and resonator voltage  follows a s i n e curve.  along  a v o l t a g e d i s t r i b u t i o n i n the  That i s to say, we  the a c c e l e r a t i n g gap but  Q  3  might o b t a i n a  c o u l d s t i l l be  low.  I f we  c o n t r o l of the t i p l o a d i n g c a p a c i t y when d e f l e c t i n g the p a n e l s i t u a t i o n would be more f a v o u r a b l e . capacities AC^^ the a b i l i t y  Still  uniform  tips  had  a  the  the d i f f e r e n c e i n the t i p l o a d i n g  as mentioned i n i ) c o u l d cause o t h e r  troubles.  to tune the segments to the same resonance by  l o a d i n g c a p a c i t y i s b e t t e r than i n the case when the hot  I t seems t h a t  changing j u s t arm  and  the t i p  ground  t i p s are d e f l e c t e d .  Some approximate c a l c u l a t i o n s as to what c a p a c i t a n c e  r e q u i r e d can then be  done.  D u r i n g the t e s t s (see S e c t i o n I I I . 4 . 2 ) the v o l t a g e s p o i n t s were measured  [see F i g . 2 4 ( c ) ] .  to p l o t the e q u i p o t e n t i a l s . using  a l e a s t square f i t .  A p o s i t i o n i n each row  points with potential.  A reasonable  the same p o t e n t i a l and  arm  at a g i v e n g r i d  A computer program was  In the program the v o l t a g e s  corresponds to a g i v e n v o l t a g e .  full  is  of  w r i t t e n and  i n each row  are  used  fitted  i s then found which  curve i s drawn through a s e t of  t h i s curve i s i d e n t i f i e d w i t h  an  equi-  - 54 6.  BEAM LOADING If in  the i n j e c t e d p a r t i c l e s are not s y n c h r o n i z e d w i t h the RF v o l t a g e phase  the c e n t r a l r e g i o n the t o t a l v o l t a g e on c a v i t y i s b o t h l e s s than the a p p l i e d  voltage V  and s h i f t e d i n phase w i t h r e s p e c t to i t .  Vg induced by the beam of charged p a r t i c l e s . p a r t i c l e s pass The  I f we  T h i s i s due  to the v o l t a g e  l e t the beam of  charged  through an u n e x c i t e d c a v i t y the magnitude o f V^. i s v e r y s m a l l .  amount o f energy  t r a n s f e r r e d through  t h i s p r o c e s s w i l l be n e g l e c t e d .  However, i f the c a v i t y i s e x c i t e d by means o f some e x t e r n a l sources of RF energy  the beam induced v o l t a g e V,, i s e s s e n t i a l l y determined  r e s o n a t o r shunt  r e s i s t a n c e and a t o t a l c i r c u l a t i n g c u r r e n t i n the  As a r e s u l t o f t h i s beam-RF f i e l d d e l i v e r a c e r t a i n amount o f energy  6.1  through  Beam-RF F i e l d  a  cyclotron.  i n t e r a c t i o n the beam can e i t h e r absorb i n t o the e x i s t i n g RF  or  fields.  Interaction  A. The RF Fundamental O p e r a t i o n The Dee  can be r e p r e s e n t e d w i t h lumped parameters  the known energy frequency  s t o r e d , power d i s s i p a t e d , q u a l i t y f a c t o r and the  [see F i g . 2 5 ( a ) ] .  t h a t the amplitude Let  R, L, C c a l c u l a t e d resonant  L e t the power i n p u t to the Dee be c o n t r o l l e d  i s always kept c o n s t a n t at a pre-determined  the c y c l o t r o n d e l i v e r a beam I .  from  such  value.  Using the u s u a l d e f i n i t i o n of  c u r r e n t as:the r a t e of f l o w o f charge a c r o s s an i n t e r f a c e the  circulating  c u r r e n t i n the c y c l o t r o n becomes  I  c  = N  T  I  (6.2)  B  where N^ r e p r e s e n t s the number of times a p a r t i c l e s has o r b i t e d i n the f i e l d between i n j e c t i o n and e x t r a c t i o n .  magnetic  I t i s a l s o assumed t h a t t h e r e are no  beam."losses i n the c y c l o t r o n . D u r i n g the course o f a c c e l e r a t i o n the p a r t i c l e s are not u n i f o r m l y  - 55 d i s t r i b u t e d on each t u r n .  They a r e c o n c e n t r a t e d i n bunches whose number i n  each o r b i t i s found from a known harmonic amplitude o f c u r r e n t p u l s e s (bunches) I  acceleration. which  We w i l l now d e r i v e an  i n average g i v e the c u r r e n t I  A [see F i g . 2 5 ( b ) ] .  B  The c o n d i t i o n o f i s o c h r o n i s m s t a t e s t h a t an i o n completes  one f u l l o r b i t i n T_ Ion  =  ~z — f_  ~  =  Ion where f j  Q  n  (6.3)  f. 1  i s an i o n r o t a t i o n f r e q u e n c y , f  of r e s o n a t o r and h ' i s a harmonic  = h ' f - j . ^ i s a resonant f r e q u e n c y  acceleration.  A t o t a l number o f p a r t i c l e s  coming out o f c y c l o t r o n p e r u n i t time i s Ng = 6.25 I  x 10  particles/sec  1 8  (6.4)  A number o f p a r t i c l e s i n each o r b i t i s found as  N  0 "  N  B Ion T  ( 6  and the p a r t i c l e s a r e c o n c e n t r a t e d i n h " bunches.  are uniformly d i s t r i b u t e d  i n both areas we  Assuming t h a t the  find  I * = I„ 1 A B q + p  (6.6)  2  I f t h e r e a r e h/ bunches per c y c l o t r o n o r b i t time o n l y l/h'* o f the t o t a l c i r c u l a t i n g d u r i n g any h a l f c y c l e o f the RF.  then a t any one  c u r r e n t i s undergoing  acceleration  During any complete RF c y c l e 2/h' o f the  t o t a l c i r c u l a t i n g c u r r e n t i s undergoing a c c e l e r a t i o n . complete  5 )  We want to determine the  amplitude 1^ (see F i g . 25) r e p r e s e n t i n g N^/h" p a r t i c l e s . particles  '  T a k i n g i n t o account a  course o f a c c e l e r a t i o n the t o t a l c u r r e n t undergoing a c c e l e r a t i o n i s  Ij- = 2 N  where we c a l l I  T  I  (6.7)  B  an i n t e r a c t i o n c u r r e n t .  Consequently, the c u r r e n t  amplitude  - 56 r e p l a c i n g a continuous  -  c u r r e n t I^. i s d e f i n e d  S i n c e the c y c l o t r o n i s isochronous  we  as  can c a l c u l a t e  f o r any  injection  phase u s i n g  N  T  4e^sT  =  <"> 6  where Eg i s the f i n a l beam energy, V  i s the r e s o n a t o r v o l t a g e and  9  8 = cot i s  the phase angle between the maximum a c c e l e r a t i n g v o l t a g e and p a r t i c l e motion across  the a c c e l e r a t i n g gap.  We  assume t h a t the time taken  i s v a n i s h i n g l y s m a l l compared to o r b i t c r o s s the a c c e l e r a t i n g gap cyclotron before reaching  circulating period.  P a r t i c l e s which  the f i n a l energy.  Those p a r t i c l e s  coming e i t h e r  to reach the same f i n a l  d e f i n e an e f f e c t i v e v o l t a g e t h a t a c c e l e r a t e s a bunch of p a r t i c l e s Iv l(sinp..+ V  av  =|V  I <cos cot> = 1  av  =  energy. as  sinq) (6.10)  p + q  v  T h i s allows us to c a l c u l a t e an average number of turns  N  gap  at the RF v o l t a g e peak spend a minimum time i n the  e a r l i e r o r l a t e r complete more turns i n o r d e r We  to c r o s s the  as  ~ 4e V  (6.11)  av To be  c o n s i s t e n t w i t h the f a c t t h a t a bunch does c o n t a i n p a r t i c l e s  d i f f e r e n t i n j e c t i o n phases we  replace N  i n eqn.  (6.7) w i t h N  J.  .  with In  those  cW  cases where the phase w i d t h of a bunch i s v e r y s m a l l use of  would  be  justified. Let  us r e s t r i c t  current pulse  our a t t e n t i o n to one  RF  c y c l e and  ( i . e . a bunch) as shown i n F i g . 25(b)  shown i n F i g 25(b).  and  In a complex n o t a t i o n V (cot) takes R V (cot) -|V I R  D  2  assume the shape of a the RF v o l t a g e wave as the form (  6  >  1  2  )  - 57 Using Fourier cosine  a n a l y s i s we can expand beam c u r r e n t  functions  [provided  o = -r 2  that  a  f (wt)  where a  1-  f(u>t + 2TT) = £(u>t)]  °° V (a cosncot + b s i n n u t ) n n n=l  (6.13)  A(p + q) TT  =  O  n  _ A ( s i n . np + s i n nq) nu  n  = A(cos>'np - cos nq) . nu  b  p u l s e s i n terms o f s i n e and  The beam l o a d p e r one RF c y c l e i s found by e v a l u a t i n g  the i n t e g r a l  f2Tr/to P" =  V_(ait) f(u)t) d t  (6.14)  R  0 Integrating  P' by  and m u l t i p l y i n g  f  gives  the complex beam power per u n i t  time p e r one Dee P = P  2  + jP  (6.15)  2  where l  P  P The e x p r e s s i o n Substitution  2r7' l'  =  2  V  =  ( s i n p  +  I^IV^cosp  S  ±  n  q  (.6.16)  )  - cosq)  (6.17)  (6.16) r e p r e s e n t s the amount of RF power d e l i v e r e d  f o r A i n eqn. (6.16) would show t h a t  to the beam.  this expression i s consistent  w i t h the u s u a l d e f i n i t i o n o f a beam r e s i s t i v e power g i v e n by I  P  R R E  B =^T  However, we w i l l b e n e f i t  <- > 6 18  from our d e f i n i t i o n l a t e r .  of P r e p r e s e n t s a r e a c t i v e power s t o r e d  The imaginary  component  e i t h e r i n a c a p a c i t a n c e or an  - 58 -  inductance. A d e t u n i n g o f the r e s o n a t o r can be found as f o l l o w s .  I n c l u d i n g the beam  l o a d i n our Dee r e p r e s e n t a t i o n y i e l d s t h e c i r c u i t shown i n F i g . 25(c) w i t h  12  l„  (6.19)  2P 1_J 2P  (6.20)  S i n c e the r e s o n a t o r i s s t i l l  forced to o s c i l l a t e  a t f = f , the phase angle o f  the t o t a l admittance Y I R is calculated  '  l R_ B  l jtC  1  (6.21)  J  from R tan  $ =  (6.22) X (R+ B  V  A new resonant frequency i s determined  n  2CX.  B  from  (6.23) (2CX )' B  B. The RF F l a t - t o p O p e r a t i o n We w i l l now extend our a n a l y s i s to the v o l t a g e waveform d e s c r i b e d by  V„(cot) = IV Icosco-, t - |Vjcosto t 3 3 K 1 -J0  In  a complex n o t a t i o n t h i s becomes  V (cot) - | v i | e J u l t R  |v |e 3  -  j U  A l l assumptions made i n p a r t A. apply h e r e . t h i r d harmonic resonant frequency f  f  3  =• 3f 1  (6.24)  t  3  3  I n a d d i t i o n we assume t h a t the  satisfies  (6.25)  - 59 -  and that the amplitude V led external sources.  is maintained at a fixed level by means of control-  3  When calculating the amplitude A, eqn. (6.8), V  must  be modified as follows V  =|vj  av  1  <costo.t> - I v J <costo.t> l  '  3  3  - i ,  p + q T h i s v a l u e of V  av  (6.26)  IV^Ksinq + sihp) - — (sin3q + sin3p)  i s then used  f o r c a l c u l a t i o n of N  av  .  A l l quantities  and  parameters have the same meaning as b e f o r e . The  complex beam power i s determined  i n t e g r a t i o n and s u b s t i t u t i o n we p  =  with e |V / =  3  T  B  ~  v j > 0.  fundamental and  the  find ^  B  2e  _  Performing  I„E (sinp + s i n q ) + j ( c o s p - cosq)  F  p  as i n p a r t A.  £ —I (sinp + s i n q ) - -r-(sin3q + 3 B^B _e  2e  3  ^  sin3p)  (sin3p + sin3q) + j (cos3p - cos3q)  e  ( s i n p + s i n q ) - — (sin3q +  P^ and P^ are the complex beam powers f o r the  the t h i r d harmonic, r e s p e c t i v e l y .  \o.za)  sin3p)  RF  A n e g a t i v e v a l u e of a r e a l  p a r t of P^ means t h a t the beam d e l i v e r s a c e r t a i n amount of power i n t o  the  t h i r d harmonic mode. By i n s p e c t i o n of the d e f i n i t i o n o f V from  the f i r s t harmonic o f the RF.  absorbed  by  we  deduce t h a t t h i s power i s taken  However, the t o t a l amount of the RF power  the beam (both harmonics c o n s i d e r e d ) s t a y s c o n s t a n t per u n i t  The beam l o a d may  time.  thus be i n t r o d u c e d by p l a c i n g an a d d i t i o n a l n e g a t i v e  r e s i s t a n c e along w i t h an a d d i t i o n a l r e a c t a n c e i n t o the t h i r d harmonic  circuit.  The  r e a l and imaginary p a r t s of P  (6.19)  and  (6.20) i n o r d e r to c a l c u l a t e Rg,  and P Rg,  are s u b s t i t u t e d i n t o eqns. Xg, X^.  C a l c u l a t i o n of Rg,  Xp enables us to f i n d a d e t u n i n g of e i t h e r resonant 3  circuit  Rg,  X , fi  a c c o r d i n g to eqns.  - 60 (6.22) and (6.23).  A d i f f e r e n t lumped parameter r e p r e s e n t a t i o n must be used  f o r the t h i r d harmonic. A s u b r o u t i n e BEAM was w r i t t e n and a t t a c h e d The  graphs i n F i g s . 26 - 31 correspond  of t h e RF a r e p r e s e n t . s i m i l a r r e s u l t s except  t o the main RF program RESLINE.  t o a s i t u a t i o n i n which both  harmonics  O p e r a t i o n w i t h the RF fundamental o n l y would  yield  t h a t the power d e l i v e r e d t o the beam from the RF  fundamental would be constant  f o r any p o s i t i o n o f a beam p u l s e w i t h r e s p e c t t o  the RF v o l t a g e peak ( i n j e c t i o n phase).  The v o l t a g e amplitudes  are 113 kV and  13 kV f o r the fundamental and the t h i r d harmonic, r e s p e c t i v e l y . The fundamental RF frequency centre of a pulse. presents  i s s e t a t 22.66 MHz.  The p u l s e p o s i t i o n r e f e r s t o the  The phase w i d t h i s s e t t o 80 deg.  The f o l l o w i n g T a b l e X I I  the approximate amounts o f RF power r e q u i r e d t o cover the s k i n l o s s e s  i n two Dees a t a g i v e n peak v o l t a g e . To complete t h i s we note t h a t the complex beam power P^ f o r the t h i r d harmonic v o l t a g e wave s h i f t e d w i t h r e s p e c t t o the RF fundamental by a deg i s g i v e n by I E P  =  £  B B (cosa + j s i n a ) ( s i n 3 p + sin3q) - ( s i n a - j c o s a ) ( c o s 3 p - cos3q) ^  ( s i n p + s i n q ) - - j [ s i n ( 3 q + a) + s i n ( 3 p - a)]  6  (6.29) TABLE X I I Resonator power l o s s f o r a g i v e n RF v o l t a g e Peak v o l t a g e (kV)  CRM  FIRST  113  CRM  THIRD  13  MAIN  FIRST  113  MAIN  THIRD  13  amplitude Power l o s s (kW)  145 4 1250 30  - 61 6.2 Beam Induced V o l t a g e I f we a l l o w a bunched beam of charged  particles  to pass through an  u n e x c i t e d r e s o n a t o r a l l p o s s i b l e harmonic modes o f the fundamental r e s o n a t o r frequency  can be e x c i t e d .  operation. he  energy  T  Some o f them a r e u n d e s i r a b l e f o r the RF system  t r a n s f e r r e d to the r e s o n a t o r modes i s v e r y s m a l l f o r the  beam c u r r e n t s under c o n s i d e r a t i o n .  F o r t h i s reason  a c r o s s the a c c e l e r a t i n g gap produced by induced However, i f the bunched beam passes beam-RF f i e l d  i n t e r a c t i o n takes p l a c e  through  the p e r i o d i c v o l t a g e s  f i e l d s can be n e g l e c t e d . an e x c i t e d r e s o n a t o r the  (see S e c t i o n I I . 6 . 1 ) .  As a r e s u l t the  energy t r a n s f e r occurs between the e x c i t e d r e s o n a t o r mode and the component o f the beam c u r r e n t which i s i n resonance w i t h i t .  The induced v o l t a g e s  appearing  now a c r o s s the gap are s e v e r a l o r d e r s o f magnitude h i g h e r than those mentioned above.  These v o l t a g e s always oppose the p a r t i c l e motion.  c o u l d have been determined  T h e i r magnitude  i n S e c t i o n II.6.1 from a known energy  S i n c e we are a l s o i n t e r e s t e d i n the t r a n s i e n t b u i l d - u p o f these  transfer. induced  v o l t a g e s a d i f f e r e n t approach has been chosen. L e t us c o n s i d e r the RF fundamental o p e r a t i o n .  The Dee i s a g a i n  r e p r e s e n t e d w i t h i t s lumped c o n s t a n t s as a p a r a l l e l resonant power i n p u t i s such t h a t the steady The power i n p u t i s kept  constant.  s t a t e amplitude  V  circuit.  The RF  = 100 kV i s a t t a i n e d .  T h i s assumption i s made o n l y because we  want to compare the magnitude o f the v o l t a g e induced by t h e beam w i t h V  '. I n  S e c t i o n II.6.3 i t w i l l be shown t h a t i n s t e a d o f h o l d i n g the power i n p u t c o n s t a n t the t o t a l v o l t a g e i n the fundamental mode must be h e l d c o n s t a n t .  If  the bunch o f p a r t i c l e s a r r i v e s a t the time o f maximum RF v o l t a g e V„ = V . ,the R oi r  &  v o l t a g e Vg induced by the beam i n the fundamental frequency out o f phase w i t h r e s p e c t t o the fundamental v o l t a g e .  circuit  Should  i s 180 deg  the bunch not be  s y n c h r o n i z e d w i t h r e s p e c t t o the RF fundamental v o l t a g e peak the phase d i f f e r e n c e between the induced v o l t a g e and t h e RF fundamental v o l t a g e would  - 62 lie  i n the is  less  range from 90 deg  to 270  deg  [see F i g . 2 5 ( d ) ] .  than the a p p l i e d RF v o l t a g e V ^ and  i s shifted  This t o t a l  voltage  i n phase w i t h  respect  to i t . L e t us  t u r n our  1 -  I  where I , I  1 2  are  2 -  I  the RF  branch c u r r e n t s .  V  a t t e n t i o n to F i g . 2 5 ( a ) .  +  1  +  2aV  2  + w^Vj-  write  ( 6  R  I  e x t e r n a l and  beam c u r r e n t  generators  components l e a d s  and  '  3 0 )  I , I , I are L L K  to  £  (6.31)  2RC  1=1," i s d e f i n e d by  2  (6.13),  = I  1^  i s given  We  cos wt  (6.32)  assume t h a t the RF  e x t e r n a l generator  L a t e r when the steady s t a t e RF  the beam c u r r e n t  generator  S u b s t i t u t i o n of L a p l a c e  V  a current  I  t r a n s f o r m s i n eqn.  =  U J  +  2  I n=l  b  C(s  ) ( S  2  2as  e  n u ) s  2  +  to ) 2  1  + 2as + t o )  2  2  . . 2 2  a  n  n  C(s  +  2  2  a  been reached  at t = t , .  (6.31) r e s u l t s i n  -ti s  n  at  (6.33)  + n to )(s  2  _ _ £ n=l  2  i s s w i t c h e d on  i s switched on  2  C(s  c  with  I  fundamental v o l t a g e has  (s) = A + B + C  A =  B  by  o  1  where to = t o ^ .  I  eqn.  I,  t = 0.  C  I  can  ^ a =  2  +  S u b s t i t u t i o n f o r current  with  I  L  I  We  to  e  + n to )(s 2  2  2  -t,s i + 2as + t o ) 2  - 63 -  V^Ct)  i s found by e v a l u a t i n g the i n v e r s e L a p l a c e t r a n s f o r m s o f eqn. (6.33).  A i s r e s p o n s i b l e f o r a p e r i o d i c v o l t a g e on the c a v i t y which, i n f a c t , accelerates particles.  Only those components i n B, C, whose frequency  c o i n c i d e s w i t h u , i . e . n = 1, have an e f f e c t on the magnitude o f the beam induced v o l t a g e V,,. operation.  The same procedure  c o u l d be a p p l i e d t o the RF f l a t - t o p  The r e s u l t a n t equations f o r V  would be those g i v e n by eqns.  3  (6.31)  and  (6.33) except f o r an o p p o s i t e s i g n o f I  The  t o t a l v o l t a g e on t h e c a v i t y i s then a sum of the t o t a l v o l t a g e s i n e i t h e r  harmonic mode, i . e . a sum o f V „, V„ , V ol B^ 0  3  3  and d i f f e r e n t v a l u e s of a, R, C,L.  , V„ . B 3  A s i m p l e computer program was w r i t t e n to determine b o t h the t r a n s i e n t and the steady s t a t e response o f i n d i v i d u a l components of a beam induced v o l t a g e . The  r e s u l t s p l o t t e d i n graphs  i n F i g s . 32 - 36 show a t r a n s i e n t  response b o t h i n the fundamental  and the t h i r d harmonic modes when the beam  c u r r e n t g e n e r a t o r has been s w i t c h e d on.  The graphs  the s i t u a t i o n i n which the RF fundamental ions.  t o t a l voltage  i n F i g . 32 c o r r e s p o n d t o  o n l y i s used  f o r acceleration of  Beam induced v o l t a g e s d u r i n g the RF f l a t - t o p o p e r a t i o n a r e shown i n  F i g s . 33, 34.  The steady s t a t e components o f a beam induced v o l t a g e a r e  p r e s e n t e d i n the f o l l o w i n g graphs.  F i g . 35(a) shows the a p p l i e d  fundamental  RF v o l t a g e which can be compared w i t h the v o l t a g e induced by the beam i n the f i r s t harmonic c i r c u i t or l a t e r , on c a v i t y .  shown i n F i g . 3 5 ( b ) .  I f the beam a r r i v e s e i t h e r  sooner  the induced v o l t a g e i s s h i f t e d i n phase and so i s a t o t a l v o l t a g e The steady s t a t e components o f a beam induced v o l t a g e when the RF  f l a t - t o p o p e r a t i o n i s c o n s i d e r e d a r e p r e s e n t e d i n F i g . 36. s c a l e i s used  i n F i g . 36(b).  time  A t o t a l v o l t a g e on c a v i t y i s a sum o f a v o l t a g e  waveform due to e x t e r n a l RF g e n e r a t o r s v o l t a g e s i n b o t h the f i r s t  A different  [see F i g . 35(a)] and beam induced  and t h i r d harmonic c i r c u i t s .  The parameters  i n the c a l c u l a t i o n can be found i n Appendices  A, B, C.  amplitudes  = 100 kV o r V  due t o e x t e r n a l g e n e r a t o r s were V  The steady  used  state  = 113 kV and  - 64 -  V  q 3  = 13 kV f o r the fundamental or the RF f l a t - t o p o p e r a t i o n , r e s p e c t i v e l y .  6.3  Concluding  Remarks  Due to the assumptions made (shape o f a beam p u l s e , f i r s t harmonic frequency  to t h i r d  r a t i o , no beam l o s s e s i n the c y c l o t r o n e t c . ) the r e s u l t s  r e p r e s e n t an i d e a l s i t u a t i o n which w i l l h a r d l y e x i s t i n the c y c l o t r o n . S i n c e the beam p u l s e possesses becomes v e r y important peak.  a c e r t a i n phase spread,  i n c e n t e r i n g the bunches w i t h r e s p e c t to the RF v o l t a g e  The c e n t r e phase can e a s i l y be found  f o r any beam p u l s e which d i s p l a y s  a symmetry w i t h r e s p e c t t o some r e f e r e n c e phase. p u l s e does not possess  the c e n t r e phase  However, even i f the beam  any symmetry i t i s always p o s s i b l e t o f i n d a p o s i t i o n o f  the p u l s e w i t h r e s p e c t to the RF v o l t a g e peak which would r e s u l t i n a r e s i s t i v e beam, l o a d  (no frequency  detuning).  Throughout S e c t i o n II.6.2 i t was assumed t h a t a steady i n t e r a c t i o n c u r r e n t was suddenly the t r a n s i e n t response  from the r e a l s i t u a t i o n .  However, the time  a c h i e v e a steady s t a t e v a l u e r e p r e s e n t s approximately  would e v e r be needed.  For t h i s  reason  o f the t o t a l v o l t a g e i n e i t h e r harmonic mode shown i n  F i g s . 32 - 34 i s d i f f e r e n t to  i n t r o d u c e d i n t o the c a v i t y .  s t a t e value of  The assumption;  from the e x t e r n a l g e n e r a t o r  i s constant  on the r e s o n a t o r w i t h and w i t h o u t  the l e a s t  i t takes  time t h a t  i n S e c t i o n II.6.2 t h a t the power output allowed us t o compare the t o t a l v o l t a g e s  a beam.  U n f o r t u n a t e l y t h i s assumption would  l e a d t o the dependence o f the t o t a l i n t e r a c t i o n c u r r e n t on the f u t u r e v a l u e s o f the a c c e l e r a t i n g v o l t a g e .  I n s t e a d , as i n S e c t i o n II.6.1 we must assume t h a t  the t o t a l v o l t a g e i s always kept voltage Let  constant  and i s always equal t o the i n i t i a l RF  (without a beam). us c o n s i d e r the RF fundamental o p e r a t i o n .  The r e s o n a t o r i s matched to  the power a m p l i f i e r o p e r a t i n g r e s i s t a n c e by means of a resonant steady  s t a t e RF v o l t a g e has been a t t a i n e d .  line.  The  The beam i s then i n j e c t e d e i t h e r a t  - 65  a full  c u r r e n t or a f r a c t i o n of i t and  the a c c e l e r a t i o n s t a r t s .  t h a t the i n j e c t e d beam i s c e n t e r e d w i t h beam l o a d can be r e p r e s e n t e d  respect  I f we  The  assume  to the RF v o l t a g e peak,  as an a d d i t i o n a l r e s i s t a n c e i n our  parameter r e p r e s e n t a t i o n of a Dee. tend  -  the  lumped  t o t a l v o l t a g e on the r e s o n a t o r would  to drop because of the beam induced  voltage.  The  amount of the RF power  d e l i v e r e d to the Dees must i n c r e a s e to compensate f o r the power e x t r a c t e d the beam d u r i n g resonator  the course  i s always kept c o n s t a n t .  network, which i n our must be  of the a c c e l e r a t i o n .  by  Thus, the t o t a l v o l t a g e on  the  Moreover the parameters of the matching  case c o n s i s t s of a resonant  l i n e with  three c a p a c i t o r s ,  r e a d j u s t e d because the shunt r e s i s t a n c e of the r e s o n a t o r  i s altered.  Computer r e s u l t s show t h a t the phase angle of the impedance at the i n p u t of the t r a n s m i s s i o n l i n e i n c r e a s e s from zero to l e s s than two  deg when the  r e s i s t i v e beam l o a d of 300  However, the  kW  i s removed  (or v i c e v e r s a ) .  magnitude of the i n p u t impedance i n c r e a s e s from 50ft to about 65  ft.  For the  CR  c y c l o t r o n the e f f e c t o f the beam l o a d on the RF system i s n e g l i g i b l e . If not  the beam i s c e n t e r e d no detuning  centered  resonator  the a c c e l e r a t i o n can s t i l l  resonant  must i n c r e a s e . frequency  frequency.  Since  takes p l a c e .  take p l a c e d e s p i t e the  As b e f o r e ,  the r e s o n a t o r  the r e s u l t s of our  shifted  the power i n p u t to the  i s f o r c e d to o s c i l l a t e at the  the t o t a l v o l t a g e waveform i s not  phase s h i f t s  However, i f the beam i s  an exact  c a l c u l a t i o n are s t i l l  L e t us t u r n our a t t e n t i o n to the RF  flat-top  s i n e wave.  resonator original For  small  valid.  operation.  The  resonator  matched b o t h to the o p e r a t i n g r e s i s t a n c e o f the fundamental a m p l i f i e r and the o p e r a t i n g r e s i s t a n c e of the t h i r d harmonic a m p l i f i e r by means of separate  transmission l i n e s .  r e s p e c t to the RF two  The  beam i s now  fundamental v o l t a g e peak.  components - one w i t h  i n j e c t e d and  The beam induced  the fundamental mode frequency  the t h i r d harmonic mode frequency.  The  centered  and  is  to  two with  v o l t a g e now  has  the o t h e r one  fundamental component of a beam  with  induced  - 66 v o l t a g e i s 180 deg out of phase w i t h r e s p e c t to the o r i g i n a l fundamental RF voltage.  The t h i r d harmonic component of the beam induced  i n phase w i t h  the o r i g i n a l t h i r d harmonic RF v o l t a g e .  t o t a l t h i r d harmonic v o l t a g e would tend  to i n c r e a s e .  t r a n s m i s s i o n l i n e must be s l i g h t l y r e a d j u s t e d i n o r d e r match.  Similarly  v o l t a g e i s , however,  T h i s i m p l i e s t h a t the The f i r s t  to a r r i v e a t a proper  the fundamental RF a m p l i f i e r must supply  of power which i s r e q u i r e d i n order  harmonic  an a d d i t i o n a l amount  to a c c e l e r a t e a beam I  to the f i n a l energy E .  Jt> But  t h i s time the fundamental RF a m p l i f i e r must a l s o s u p p l y  amount o f power which i s absorbed by the beam.  O  a second a d d i t i o n a l  However, t h i s power i s not used  f o r a c c e l e r a t i o n o f p a r t i c l e s , but i t i s d e l i v e r e d by the beam i n t o the t h i r d harmonic mode.  At h i g h c i r c u l a t i n g c u r r e n t s the amount of power the beam  d e l i v e r s i n t o the t h i r d harmonic mode becomes a s i g n i f i c a n t p o r t i o n o f the t o t a l RF power r e q u i r e d to cover s k i n l o s s e s i n the t h i r d harmonic mode.  The shunt  r e s i s t a n c e o f a Dee o s c i l l a t i n g i n the t h i r d harmonic mode i s e s s e n t i a l l y i n f l u e n c e d by the presence o f the r e s i s t i v e beam l o a d . t r a n s m i s s i o n l i n e parameters i n o r d e r r e s i s t a n c e becomes more d i f f i c u l t .  Retuning the  to match an a l t e r e d r e s o n a t o r  Besides  shunt  the t r a n s m i s s i o n l i n e r e t u n i n g the  power from the t h i r d harmonic a m p l i f i e r must e i t h e r be reduced o r some p a r t o f the t h i r d harmonic output i n order  to m a i n t a i n  power must damped  i n a dummy l o a d .  T h i s must be done  the t h i r d harmonic v o l t a g e peak c o n s t a n t .  power which s h o u l d be damped  i s equal  beam i n t o the t h i r d harmonic mode.  The amount o f  to the amount o f power d e l i v e r e d by the  The power d e l i v e r e d i n t o the t h i r d  harmonic  mode i n the CR c y c l o t r o n i s v e r y s m a l l and the computer r e s u l t s show t h a t the i n p u t impedance i s o n l y s l i g h t l y a f f e c t e d . I f the i n j e c t e d beam i s not c e n t e r e d , harmonic mode o c c u r s . resonance o c c u r s .  Since  a frequency  the f r e q u e n c i e s  detuning  of e i t h e r  are f i x e d a phase s h i f t  from the  The r e s u l t a n t v o l t a g e waveform i n the fundamental mode i s  o n l y s l i g h t l y d i s t o r t e d because the frequency  detuning  i s small.  However, the  - 67 t o t a l v o l t a g e waveform i n the t h i r d harmonic mode i s s e r i o u s l y  distorted  because the frequency d e t u n i n g i s s e v e r a l times l a r g e r than t h a t i n the f i r s t harmonic mode. Af  i s negative.  presence  T h i s means t h a t the r e s u l t a n t  o f the beam q u a l i t y and e v e n t u a l l y t o a complete l o s s o f the beam.  t o the fundamental v o l t a g e wave.  i s changed.  with  resonant  The r e s u l t a n t v o l t a g e f l a t - t o p wave i s a g a i n d i s t o r t e d  by the t h i r d harmonic component o f the beam induced v o l t a g e . the subsequent b e h a v i o u r automatic  shifted  I f the c e n t e r e d beam w i t h r e s p e c t to  the fundamental v o l t a g e i s b e i n g a c c e l e r a t e d the t h i r d harmonic frequency  i s positive,  This process leads to  c o u l d a l s o happen t h a t the t h i r d harmonic v o l t a g e i s i n i t i a l l y  respect  1  f l a t - t o p v o l t a g e wave w i t h the  of the beam i s even more s e r i o u s l y d i s t o r t e d .  a worsening It  I t i s a l s o c l e a r from F i g s . 26 - 31 t h a t w h i l e A f  c o n t r o l system.  In e i t h e r  case,  o f the c y c l o t r o n depends on the c a p a b i l i t y o f the  - 68 7.  COUPLING LOOP ASSEMBLY 7.1  Power Loss i n the Vacuum U n l i k e the r e s o n a t o r  resonant  Seal  system, which i s p l a c e d i n a vacuum chamber, the  t r a n s m i s s i o n l i n e i s operated  in air.  F o r t h i s reason  a transition  from vacuum t o a i r has to be employed somewhere near the c o u p l i n g l o o p . vacuum feedthrough s h o u l d i)  serve  This  two purposes:  g i v e a vacuum-tight s e p a r a t i o n between the main vacuum tank and the t r a n s m i s s i o n l i n e and  ii) The  help maintain  t h e c o r r e c t p o s i t i o n o f the l o o p .  s e a l i s made i n a form of a . c i r c u l a r d i s c as shown i n F i g . 37.  d i s c i s s t r e s s e d by m e c h a n i c a l l o a d s shrinkage  The  ( f o r c e s g i v e n from the Dee, by the  o f the d i s c a t the assembly and the d i f f e r e n c e o f p r e s s u r e )  and  thermal s t r e s s e s o r i g i n a t e d from a non-uniform temperature i n t h e body o f the d i s c and a r e s t r a i n e d d i l a t i o n a l o n g The  the edges.  power l o s s i n d i e l e c t r i c m a t e r i a l i s c a l c u l a t e d from  P  2  1  (7.1)  2Rp  where (7.2)  1 tan 6  Q  The  capacitance  (7.3)  between two c i r c u l a r conductors i s found as 2 i r  C  =  £  r o £  t  (7.4)  lnoyiy The  t o t a l power l o s s i n t h e d i s c i s determined from  (7.5) '2  1  where t i s the t h i c k n e s s o f t h e d i s c , R  are o u t e r  and i n n e r r a d i i , V i s  - 69 the v o l t a g e d i f f e r e n c e a c r o s s the d i s c , of the d i l e c t r i c m a t e r i a l , e  i s the r e l a t i v e d i e l e c t r i c c o n s t a n t  i s the r e l a t i v e d i e l e c t r i c c o n s t a n t i n t h e  o  vacuum, t a n 5 i s the l o s s tangent., f i s the o p e r a t i n g frequency and a i s the thermal c o n d u c t i v i t y . tan 6 = 0.0002. lvi=  F o r the d i s c made up from ceramic we have  T a k i n g 2R  = 6.5 i n . , 2R, = 1.5 i n . , f = 22.66 MHz,  2  9.1 kV (CRM), |V|= 12.7 kV (main c y c l o t r o n ) ,  i n t o eqn. (7.5) r e s u l t s i n P  t = 0.5 i n . and s u b s t i t u t i n g  = 5.1 tf. U s i n g the d i s c w i t h the same dimensions  i n the main c y c l o t r o n would g i v e P  = 10 W.  Taking t e f l o n m a t e r i a l with  t a n 6 -0.0005 (most p e s s i m i s t i c v a l u e ) and e r However, i t s h o u l d be understood different  = 2.1 we g e t f o r CRM P 1  = 2.9 W.  &  t h a t h i g h power o p e r a t i o n a t a frequency  from the r e s o n a t o r resonant f r e q u e n c y might  l o s s i n the vacuum s e a l .  = 9 and  l e a d t o a much h i g h e r RF  I t i s shown i n S e c t i o n II.10.3 t h a t the r a t i o o f  v o l t a g e a t the r e s o n a t o r t i p t o v o l t a g e a t l o o p , V../V , depends on the operating frequency.  F o r lower v a l u e s o f t h i s r a t i o  on r e s o n a t o r t i p s can r e s u l t  i n a v o l t a g e a c r o s s the d i s c which i s i n c r e a s e d  by a f a c t o r of two over the computed v a l u e . amount o f RF power l o s s 7.2  the o p e r a t i o n w i t h 100 kV  I t i s then obvious t h a t the  increases.  Power Loss i n A d j a c e n t Areas. The d i s c i s u s u a l l y covered w i t h s i l v e r on b o t h t h e i n n e r and o u t e r edges  to y i e l d a good and r e l i a b l e vacuum-tight w i l l be l o s t  A c e r t a i n amount o f power  i n these s u r f a c e s , which w i l l produce  vacuum f e e d t h r o u g h .  a d d i t i o n a l h e a t i n g o f the  The power l o s s i n the c i r c u l a r conductor i s found 1  P  where p  seal.  t  P  2  c ; |  2 - I - 2 ^ '  1  '  (  i s a surface r e s i s t i v i t y , p S  = 2.6/f x 10  7  for silver,  S  the c u r r e n t through l o o p ( i n p u t c u r r e n t ) . the o u t e r conductor and P  from  7  and I - I  -  6  is J-i  So we have f o r CRM P^ = 0.01 W i n  = 0.05 W i n the i n n e r conductor.  )  Power l o s s e s i n  - 70 the t r a n s m i s s i o n l i n e conductors above and below the ceramic d i s c a r e calculated according yields P  eqn. (7.6), where we take t = 23.5 cm.  .= 0.17 W f o r the o u t e r conductor and P  3  conductor.  3  A substitution  = 0.44 W f o r the i n n e r  S i n c e the c u r r e n t through the loop i n the main c y c l o t r o n i s t e n  times l a r g e r , the v a l u e s approximately  o f P^ and P^ would be i n c r e a s e d by a f a c t o r o f  100 p r o v i d e d  t h a t the same dimensions a r e used f o r the main  cyclotron. Once a g a i n the v a l u e s operating  7.3  frequency  o f P^ and P^ a r e s i g n i f i c a n t l y i n f l u e n c e d i f the  i s different  from the r e s o n a t o r  frequency.  Power L o s s i n the C o u p l i n g Loop The  power l o s s i n the c o u p l i n g loop i t s e l f  i s produced by the i n p u t  c u r r e n t and by the c i r c u l a t i n g c u r r e n t i n the r e s o n a t o r .  The i n p u t c u r r e n t i s ,  i n f a c t , r e s p o n s i b l e f o r the d e l i v e r y o f the r e q u i r e d amount o f power t o the Dees. loop  T h i s i n p u t c u r r e n t i s spread  over most of the lower p a r t o f the c o u p l i n g  ( t h a t p a r t o f the loop f a c i n g the ground arm).  T h i s f a c t must be taken  i n t o account when computing the power l o s s due to the i n p u t  current.  Let %' be the l e n g t h and w' be the w i d t h o f the c o u p l i n g l o o p .  The power  l o s s due t o the i n p u t c u r r e n t i s then c a l c u l a t e d from  P, = ^ R U J k z s L where R  s  = p  (7.7)  2  %" —- i s the s u r f a c e r e s i s t a n c e and I i s the loop c u r r e n t . s w L T  a l s o know t h a t a t a g i v e n p o i n t i n the r e s o n a t o r constant  i n the v e r t i c a l d i r e c t i o n .  magnetic f i e l d the f i e l d  l i n e s are d i s t o r t e d .  the magnetic f i e l d i s  I f the c o u p l i n g loop i s i n s e r t e d the As a f i r s t  a p p r o x i m a t i o n we assume t h a t  d e n s i t y a t the s u r f a c e o f the c o u p l i n g loop i s the same as t h a t a t  the c a v i t y w a l l . The  We  T h i s i m p l i e s t h a t t h e r e i s a l s o the same c u r r e n t  power l o s s due to the r e s o n a t o r  c i r c u l a t i n g c u r r e n t i s found as  density.  (7.8) where I  w  i s the c u r r e n t i n a s t r i p of the r e s o n a t o r w a l l .  s t r i p i s e q u a l to the loop w i d t h .  Both P^  and P  5  The w i d t h of  the  are c a l c u l a t e d by the code  RESLINE. The resonant of  o p e r a t i o n of the system a t a frequency frequency would a f f e c t  the r e s o n a t o r shunt  resistive  the v a l u e of P^.  (parallel)  different  from the  S i n c e the r e a c t i v e component  impedance becomes comparable w i t h  component, the magnitude of the impedance p r e s e n t e d  a c o u p l i n g p o i n t i s decreased. is practically  constant  t h a t the magnitude of I  resonator  the  to the l i n e at  However, the r e s o n a t o r v o l t a g e step-up  i n the v i c i n i t y of the r e s o n a t o r frequency.  ratio  T h i s means  i s i n c r e a s e d which i n c r e a s e s the amount of power  d i s s i p a t e d i n the c o u p l i n g l o o p .  T h i s i n c r e a s e i n P^ may  r e a c h a few Watts  i n the CR c y c l o t r o n , but a few hundred Watts i n the main c y c l o t r o n .  k  - 72 8.  TRANSMISSION LINE The main f a c t o r s which i n f l u e n c e a c h o i c e o f the resonant l i n e parameters are the maximum power t h a t can be t r a n s m i t t e d permissible voltage.  Since  and, c o n s e q u e n t l y , t h e maximum  t h e l i n e i s operated  i n a i r the  probability for  g e t t i n g sparks due t o m o i s t u r e , sharp edges e t c . i s much h i g h e r . voltage The  The maximum  i n t h e l i n e depends m a i n l y on the l o a d connected a t t h e end o f l i n e .  maximum v o l t a g e  g r a d i e n t happens t o be near the i n n e r conductor and i s  g i v e n by  E  R  =  a  ln(R / 2  R l  )  (S- ) 1  where V i s the v o l t a g e d i f f e r e n c e between two B a s i c parameters s i m i l a r to those d e r i v e d however, s i n c e t h e l i n e i s terminated expressions  conductors. i n S e c t i o n I I . 1 c o u l d be found,  i n a general  would become q u i t e c o m p l i c a t e d .  complex impedance t h e  Therefore  the q u a l i t y f a c t o r ,  power l o s s , energy s t o r e d and o t h e r parameters a r e c a l c u l a t e d a c c u r a t e l y the computer programs RESLINE and MATCH.  B a s i c a l l y t h e power l o s s and t h e Clvl using  energy s t o r e d a r e c a l c u l a t e d by i n t e g r a t i n g the p r o d u c t s R III , L l l l , 2  (1.10) and (1.11).  The method f o r computing the lumped c o n s t a n t s  resonant l i n e i s o u t l i n e d i n S e c t i o n Any  simple  2  2  o f the  II.9.3.  c a l c u l a t i o n o f the l i n e ' s p r o p e r t i e s i s also hindered  presence of t h r e e c a p a c i t o r s which s i g n i f i c a n t l y a f f e c t p r o p e r t i e s o f the l i n e .  using  by t h e  the e l e c t r i c a l  The c a p a c i t o r near the loop h e l p s  adjust  the loading  impedance as d e s i r e d ; t h e c a p a c i t o r s i n t h e c e n t r e o f the l i n e and a t t h e tube tune t h e l i n e to a t t a i n a d e s i r e d r e s i s t i v e i n p u t impedance resonance o f the whole system). line.  The e l e c t r i c a l l i n e l e n g t h  the g i v e n v a l u e s  I n a d d i t i o n , the c e n t r e  ( i . e . to achieve  a  c a p a c i t o r shortens the  ( c a p a c i t o r s and loop i n c l u d e d )  o f c h a r a c t e r i s t i c impedance, l e n g t h and l o a d i n g  i s nA/2. F o r impedance  t h e r e always e x i s t s a combination of the t h r e e c a p a c i t o r s such t h a t t h e  - 73 r e q u i r e d i n p u t impedance i s o b t a i n e d . A resonant  l i n e i s d e s i r e d when d e a l i n g w i t h an u n s t a b l e  reduces the e f f e c t ,  l o a d because i t  a t the tube, o f an a l t e r e d l i n e t e r m i n a t i n g  load.  Consequently, the tube parameters may have t o be r e a d j u s t e d o n l y s l i g h t l y . a d d i t i o n , sparkovers e x i s t with resonator  i n the r e s o n a t o r s  a non-resonant l i n e .  do n o t p r e s e n t  Such a r c - o v e r s  the problems which would  cause a d e t u n i n g  of the  and, t h e r e f o r e , the system i s immediately mismatched, i . e . a match  between the tube o p e r a t i n g r e s i s t a n c e and the r e s o n a t o r longer e x i s t s .  Hence, under these  energy may reach As  In  shunt r e s i s t a n c e no  c o n d i t i o n s , o n l y a f r a c t i o n o f the r e s o n a t o r  the power tube.  i t i s a n t i c i p a t e d t h a t the r e s o n a t o r  q u a l i t y f a c t o r w i l l be s l i g h t l y  lower than the t h e o r e t i c a l v a l u e an i n v e s t i g a t i o n was c a r r i e d out to e s t i m a t e how much t h i s would a f f e c t  tube parameters.  Using  the computer program  RESLINE the system was f i r s t matched t a k i n g i n t o account a t h e o r e t i c a l v a l u e o f the Q.  The parameters were then f i x e d and the i n p u t impedance was computed f o r  differing  values  o f the r e s o n a t o r  quality factor.  i n p u t impedance v a r i e d f o r the v a l u e s the phase angle  The r e s u l t s show t h a t the  o f Q from 7000 down to 4000.  of the i n p u t impedance was always c l o s e t o zero.  c o n c l u s i o n i s t h a t the parameters o f the l i n e must be r e a d j u s t e d a m p l i f i e r i s to operate  into a given r e s i s t a n c e .  However,  The i f the power  I t was a l s o n o t e d t h a t the  v o l t a g e a t the i n p u t o f the l i n e was almost u n a f f e c t e d by the change i n Q. The  computer program RESLINE was a l s o a p p l i e d when the r e l a t i o n between  the r e s o n a t o r peak v o l t a g e and the tube parameters was i n v e s t i g a t e d . system was f i r s t matched f o r the r e s o n a t o r 100  The  fundamental v o l t a g e amplitude o f  kV and t h e l i n e parameters were then f i x e d .  I t was found t h a t f o r up t o  a 10% change t h e r e was a l i n e a r r e l a t i o n s h i p between the r e s o n a t o r fundamental voltage-, peak l i n e input.  and  the v o l t a g e and c u r r e n t a t the t r a n s m i s s i o n  The change i n phase angle o f the i n p u t impedance was n e g l i g i b l e .  - 74 The  r e s u l t s o f the computer program RESLINE r e f e r t o the o p e r a t i o n o f the  system a t the r e s o n a t o r resonant  frequency  f .  Should  r e s i s t i v e i n p u t impedance) be a t t a i n e d a t f =f f may be e i t h e r h i g h e r o r lower  a proper match ( i . e . a  the power l o s s i n the l i n e  than f o r the case f = f .  Computer r e s u l t s show  t h a t i f the i n i t i a l match i s o b t a i n e d at f = 22.652 MHz < f = 22.660 MHz the o v o l t a g e a c r o s s CP(2) i s 14.6 kV when the r e s o n a t o r t i p v o l t a g e i s 100 kV (CRM). Should we match the l i n e a t f = 22.668 MHz > f CP(2)  would be 2.5 kV.  increasing Z . o  = 22.660 MHz  The v o l t a g e i n the f i r s t  across  s e c t i o n i n c r e a s e s w i t h an  The c a p a c i t o r v o l t a g e a t f = f i s 8.4 kV. o  c y c l o t r o n the o p e r a t i o n a t f = 23.096 MHz < f  the v o l t a g e  In t h e main  = 23.100 MHz would r e s u l t i n  the v o l t a g e a c r o s s CP(2) o f about 34 kV w h i l e the nominal v a l u e a t f = f i s o 13.1 kV.  - 75 -  LUMPED PARAMETER REPRESENTATION 9.1  Resonator Lumped To enable  Constants  us t o c a l c u l a t e such  q u a n t i t i e s as the r a t e o f r i s e  of resonator  v o l t a g e , beam l o a d i n g , r e s o n a t o r v o l t a g e and i n p u t impedance as f u n c t i o n s o f generator  frequency  the r e s o n a t o r had t o be r e p r e s e n t e d  parameters R, L, C. electric Let  Their d e f i n i t i o n  f o l l o w s immediately  i n terms o f lumped once t h e magnetic and  e n e r g i e s s t o r e d i n the r e s o n a t o r have been determined. the r e s o n a t o r v o l t a g e peak be V,.  r e s o n a t o r made up o f n segments i s f . Q  the r e s o n a t o r peak v o l t a g e V  w  E = i  c  l  v  i  |  The resonant  frequency  of a  The lumped c a p a c i t a n c e i s r e l a t e d t o  by  (9.1)  2  so t h a t sin  c - f c  -2  £ +  sin  (9.2)  c  where C' i s a c a p a c i t a n c e p e r u n i t l e n g t h o f a r e s o n a t o r segment and V, = V os i n — 1 c^ S L .  The lumped i n d u c t a n c e  i s then c a l c u l a t e d  1  L =  from  (9.3)  wo C 2  Similarly  the lumped i n d u c t a n c e  i s related  to the r e s o n a t o r peak c u r r e n t I  by  (9.4) This results i n sin  2 Consequently,  £ +  n  2—V c_  (9.5)  2 ^ c  the lumped c a p a c i t a n c e i s g i v e n by 1 C =  (9.6) %  L  - 76 A l l our c a l c u l a t i o n s use lumped c o n s t a n t s L, C as d e f i n e d by eqns. (9.2) and (9.3). L, C parameters a r e u n a f f e c t e d by a c h o i c e o f r e p r e s e n t a t i o n .  However,  a s e r i e s r e s i s t a n c e must be used when the r e s o n a t o r i s r e p r e s e n t e d w i t h a s e r i e s resonant  c i r c u i t and a p a r a l l e l r e s i s t a n c e i s t o be used when r e p l a c i n g  the r e s o n a t o r w i t h a p a r a l l e l  resonant  circuit.  The p a r a l l e l r e s i s t a n c e  is  d e f i n e d by  h-^-u which y i e l d s a f t e r  (9  7)  substitution 2o« w Z - °  sin  2  R  -  2  ^ %  3  (9.8)  C  2_e. c The  s e r i e s r e s i s t a n c e R i s d e f i n e d i n a s i m i l a r way by  A l l lumped c o n s t a n t s a r e computed by the program RESLINE.  9.2  Representation of the Coupling The  network.  RF energy must be s u p p l i e d to the Dees by means o f some c o u p l i n g The most common c o u p l i n g i s made by means of e i t h e r a c a p a c i t o r or a  coupling loop.  We s h a l l r e s t r i c t o u r s e l v e s to i n d u c t i v e c o u p l i n g by means o f  a coupling loop.  L e t us c o n s i d e r loop c o u p l i n g to the resonant  shown i n F i g . 3 8 ( a ) .  The r e s o n a t o r i s r e p r e s e n t e d by a s e r i e s  c a v i t y as resonant  c i r c u i t w i t h lumped c o n s t a n t s R, L, C r e l a t e d t o the r e s o n a t o r peak v o l t a g e V (V^ = V ^ s i n cot).  L ^ and R^ are the loop s e l f - i n d u c t a n c e and the l o o p  resistance, respectively.  I ' i s t h e c u r r e n t i n the resonant  circuit, V i s  K the v o l t a g e induced  i n the loop  (jcoL  Li  I Li  i s not i n c l u d e d ) .  M = kVL L L>  L i s the  - 77 mutual i n d u c t a n c e , k/ i s the c o u p l i n g c o e f f i c i e n t ,  k i s the r e s o n a t o r  step-up  ratio. The  impedance Z "  seen a t the i n p u t t e r m i n a l s BB"  is  (9.10)  where the t h i r d  term i s i d e n t i c a l  impedance must now Two  methods may  w i t h Z'  g i v e n by eqn.  (10.18).  This  be matched by some means to the power tube stage  be employed f o r t h i s purpose.  impedance.  F i r s t when a non-resonant  line  i s used, the v a l u e of the mutual i n d u c t a n c e must be chosen so t h a t the resistive  p a r t of Z"'  i s e q u a l to the c h a r a c t e r i s t i c  At the same time a c a p a c i t i v e component has  impedance of the  line.  to be i n t r o d u c e d at the loop i n  o r d e r to c a n c e l the i n d u c t i v e r e a c t a n c e of the l o o p . The  o t h e r method, d e s c r i b e d i n S e c t i o n II.10.1, makes use of a  l i n e w i t h parameters chosen so t h a t the l i n e merely transforms Z  i n t o the tube impedance.  presents  a pure r e s i s t i v e  At the resonant  frequency,  l o a d a t the c o u p l i n g p o i n t .  c o n s i s t s o f the r e s o n a t o r w i t h a c o u p l i n g loop and tuned  to the resonant  unloaded.  frequency  the impedance  f , the  The  resonant  resonator  system which  a resonant  l i n e must be  at which the r e s o n a t o r o s c i l l a t e s  i f left  As shown i n S e c t i o n 11.10.2 the whole system i s then resonant  the r e s o n a t o r frequency  f .  Only then can a power t r a n s f e r and  at  a resonator  v o l t a g e be a c h i e v e d w i t h t r a n s m i s s i o n l i n e parameters as computed by  the  program RESLINE. At resonance f = f  o  and we  L  have  (9.11)  and dl M  dt  (9.12)  - 78 so that M = L  (9.13)  k r  The same d e f i n i t i o n of M follows from the equations of a shunt ( p a r a l l e l ) resonator resistance.  At resonance the shunt resistance seen at a coupling  point i s (9.14)  2P where P i s the resonator power l o s s .  From eqn. (9.10) R^ must also be equal to  2 2 M to  o  (9.15)  R This leads to  M = L  R This d e f i n i t i o n of M i s consistent with calculation  of the voltage V  induced Li  i n the coupling loop.  The c o e f f i c i e n t k' can be expressed as  (9.16) R In the program RESLINE, the mutual inductance i s computed according to  M  b  3  vr-  =  y  nw  (9.17) 0  °  where a, b are loop dimensions, w  i s the average width of the segment, n i s a 7  the number of segments, U  Q  = 4TT X 10  H/m.  computing i s done i n terms of distributed  One must bear i n mind that a l l parameters.  For this reason the  current through the root i s given approximately by IJ = I  where I , Q  V  q  =  (9.18)  are voltage and current peaks i n the l i n e which i s a quarter  wavelength long and V  = V sin ^ i . Q  Assuming now that the resonator has been  -  79  -  r e p l a c e d w i t h i t s lumped parameters R, L, C where  L = io C 2  D  we  f i n d t h a t the c u r r e n t i n the resonant  circuit is  (9.19)  This i s d i f f e r e n t The  two  from I  expressions  f o r a mutual i n d u c t a n c e  are r e l a t e d  to one  another  by  the f o l l o w i n g formula I M = M  av  (9.20)  R where I  av  i s the average c u r r e n t f l o w i n g i n the c a v i t y a l o n g  (the c u r r e n t which i s taken  f o r computing the l o o p induced v o l t a g e V )  L e t us r e t u r n a g a i n to a diagram shown i n F i g . 3 8 ( a ) . equations  to t h i s system we  I  =  L  find  the c u r r e n t through  R  V L  Z Z D  the c o u p l i n g loop  the loop i s  (9.21)  + w M 2  T  Applying Kirchhoff's  2  where Z  Z  which f o r f = f  L  o  = R + 3 u)L -  R  =  *L  +  W  reduces  L  L  to R  I, = V. L L Consequently,  J  Ceo  RR L  + to M o 2  the phase s h i f t  I  + ico  LR o L  (9.22)  and V' i s Lt  co L o L  2  Li  T  tan $  = 1  S i m i l a r l y we  obtain I  R  R  L  +  B  (9.23) P  - 80 jioM R  ZZ  L  + 03 M 2  L K  = - V  L  ^R  L  + 03 M 2  2  03  03 =  Q  + j 03_L  7  o  J  (9.24)  R  o L  A phase s h i f t between V " and I " a t resonance i s T  t  a  n  $  2  _  =  _  ^  (9.25)  ° L S i n c e t h e r e s o n a t o r v o l t a g e i s g i v e n by  V v  R  _ _k_  =  jcoC  V  L  l^M  Z„Z + 03 M R L Z  2  03 =  T  J  M  C  R R + L  1 2 2 2M  OJ M  ^  03L  Q  4. + J 0 )TL „ R  Z  o  the phase a n g l e between V and I i s found R Li 1 t a n  0)  (9-26)  L  t o be  -  = 0  =  (9.27)  03 = 03„  The  phase angle between I ' and V  i s found  as b e i n g i d e n t i c a l l y  equal to  90 deg t a n <& -> «>  If  t h e Q i s low t h e resonant  considered.  9.3  (9.28)  As a r e s u l t  Lumped Constant  frequency  g i v e n by eqn. (1.7) s h o u l d be  a l l v e c t o r s would be s h i f t e d  Representation  slightly.  o f a Resonant L i n e  There a r e s e v e r a l p o s s i b l e ways o f r e p r e s e n t i n g a resonant parameters at i t s resonant wide frequency  frequency,  I n o r d e r t o r e p r e s e n t the l i n e  range by lumped parameters, a number o f lumped  c i r c u i t s would have to be used. behaviour  f .  of the l i n e  S i n c e we a r e mainly  i n the v i c i n i t y o f f  l i n e by a s i n g l e lumped c o n s t a n t resonant  l i n e by lumped ina  constant  interested  i n the  t h e r e p r e s e n t a t i o n o f the resonant  c i r c u i t i s adequate.  - 81 L e t us  consider  the  following representation  L^ are known lumped constants  of the r e s o n a t o r  L^ are unknown lumped constants parameters M",  R^,  C^,  the l i n e  <W">,  tube v o l t a g e We  and  current V_  =  11  r e l a t e d to the  tube c u r r e n t L  and  =  Suppose we  know the  „, I_  1 j—jr~2  We  7rD  „ , and  lUrSJl  C^, The  average  the q u a l i t y f a c t o r  the r e s o n a n t frequency of each o b t a i n a lumped i n d u c t a n c e  2  as  4<W'> 2 TUBE  (9.29)  H  a lumped c a p a c i t a n c e c  TTT)  lUBJi  1 = -——  i s the same, i . e . co  R^,  the average magnetic energy s t o r e d i n  a l s o assume t h a t , i n i t i a l l y , 2  circuit  <W'>,  Dee,  C ,  i s the mutual i n d u c t a n c e .  L^ are f i x e d as f o l l o w s .  energy s t o r e d i n the l i n e  of the l i n e Q^.  [ F i g . 38(b)] where R^,  c o n s i s t i n g of one  of the l i n e , M "  electric  rl  -  as  2 = ^rV  <-> 93 0  o  The  s e r i e s r e s i s t a n c e i s found from  = ^ n r  R 2  2  O  Furthermore we  - >w  -  The  =  /  ( R  load  d i f f e r e n t values  fact  e q u a l to  3  1  )  (f = f )  as  i" V ^  same procedure f o r f i n d i n g R^,  terminating  The  "  -  5^  c o n d i t i o n f i x e s the mutual i n d u c t a n c e  M  9  2  r e q u i r e the i n p u t r e s i s t a n c e to match to be  h This  (  (resonator) r e s u l t not  ( 5  C^,  c o n s i s t s of two  L^, M"'  i s a p p l i e d i f the  Dees.  I t i s to'be noted  o n l y f o r the parameters of the l i n e but  t h a t the l i n e has  an i n i t i a l  frequency f  i s due  -  3 3 )  that  also for  to the  equality  - 82  -  of the e l e c t r i c and magnetic e n e r g i e s i n the l i n e when the loop inductance C , 2  9.4  L , 2  i s taken to be p a r t o f the l i n e  R,  M"  2  (see Appendix A ) .  self-  The  parameters  are computed by the program RESLINE.  Representation S i n c e i t was  of the Whole System i n Terms of Lumped Parameters  d e s i r e d to know the b e h a v i o u r  of r e s o n a t o r parameters i . e .  the r e s o n a t o r v o l t a g e , phase, power l o s s e t c . o n l y i n the v i c i n i t y resonance,  the two  parameters.  Dees and  the resonant  l i n e were r e p r e s e n t e d w i t h  A lumped parameter r e p r e s e n t a t i o n was  c a l c u l a t i n g the t r a n s i e n t response  the  is valid The  direct  IV.1.8).  r e p r e s e n t a t i o n of the system i s shown i n F i g . 43(b).  m a t r i x c o u l d be o b t a i n e d i n a s i m i l a r way  i n d i v i d u a l v o l t a g e s and  The  as i n S e c t i o n II.10.1.  s o l v e d by w r i t i n g down the K i r c h h o f f ' s equations  impedance was  and  c u r r e n t s which were of p a r t i c u l a r i n t e r e s t .  the v o l t a g e s and  c u r r e n t s computed.  The  then v a r i e d and  and  the r e s o n a t o r v o l t a g e as a f u n c t i o n of the g e n e r a t o r frequency  The  i n p u t impedance are p l o t t e d  (CRM).  i n v e s t i g a t i o n t h a t has been c a r r i e d out w i t h r e g a r d to  i n d i c a t e d t h a t the whole system possesses  resonator and  input  frequency  the  e q u i v a l e n c e o f the lumped parameter r e p r e s e n t a t i o n w i t h our r e a l system  resonant  the  calculating  was  The  transfer Instead  f i r s t matched to the tube r e s i s t a n c e , the o p e r a t i n g  i n F i g s . 39, 40  on  the program MATCH shows to what e x t e n t the r e p r e s e n t a t i o n  (see S e c t i o n  c i r c u i t was  interaction  Comparison of the r e s u l t s based  lumped parameter r e p r e s e n t a t i o n w i t h those o b t a i n e d e i t h e r from measurements o r by  lumped  a l s o n e c e s s a r y when  of the r e s o n a t o r v o l t a g e and  of the beam w i t h the c a v i t y RF f i e l d s .  of  c i r c u i t w i t h a bandwidth almost ( F i g s . 39, 40).  the tube the l o a d e d  the p r o p e r t i e s of a simple e q u a l to the bandwidth o f  S i n c e a proper match e x i s t s between the q u a l i t y f a c t o r i s e q u a l to one h a l f  For t h i s compare a g a i n the bandwidths i n F i g s . 39,  40.  has  parallel  the resonator  the unloaded  one.  - 83 10. RESONANT OPERATION OF THE RF SYSTEM 10.1  O p e r a t i o n a t the Maximum Power T r a n s f e r When matching  the resonant l o a d two c o n d i t i o n s must be s a t i s f i e d .  we have t o p r o v i d e a matching r e s o n a t o r v o l t a g e step-up  network w i t h such parameters  r a t i o can be a c h i e v e d .  a match a t a maximum power t r a n s f e r . efficiency. matching The is  that a proper  tube-  Secondly we want t o o b t a i n  The tube w i l l  then operate w i t h maximum  Two d i f f e r e n t methods e x i s t which may be a p p l i e d t o d e s i g n i n g the  network. first  one we a r e to d e a l w i t h assumes t h a t the g e n e r a t o r  tuned p r e c i s e l y t o the resonant frequency o f the r e s o n a t o r .  the r e s o n a t o r p r e s e n t s a pure r e s i s t i v e impedance.  frequency  At resonance,  T h i s shunt r e s i s t a n c e i s  to be matched to the tube o p e r a t i n g r e s i s t a n c e by means o f a resonant To s i m p l i f y t u n i n g o f the whole system,  t h r e e c a p a c i t o r s connected  p o i n t s a l o n g t h e t r a n s m i s s i o n l i n e a r e a t our d i s p o s a l . shown i n F i g . 41. parameters. and  First  line.  at various  The whole system i s  The r e s o n a t o r and the l i n e a r e c h a r a c t e r i z e d by d i s t r i b u t e d  The r e s o n a t o r v o l t a g e and c u r r e n t a r e r e l a t e d t o the tube v o l t a g e  c u r r e n t by the f o l l o w i n g s e t o f m a t r i c e s :  B T  0  =  (10.1)  cosh T  2  Zj sinh Y ^ j  YJ&J  =  (10.2) D  —sinh Y I  z  ' l  l  1  cosh Y  1 1  (10.3) j  03C  5  1  -  84  -  cosh y„l  Z  2  " 2  2  s i n h Y ft " 2  2  (10.4)  -rsinh Y ft Z 2 2  A 5  T  (10.5) D  5  R  5  j u ^  1  cosh Y £ „ 3 3  = 6•  Z  B  7  ^-sinh  y £ 3  cosh Y A '3 3  3  7 (10.7)  7  °7  A  B  0  - j CJM"  =  (10.8)  cosh Y ^ 9  10  1  h  sinh  Y l f  \  =  (10.9)  ^ i n h  A  T  sinh  = 7 . C  T  3  (10.6)  A  T  2  5  D  T  ' 2  = C  T  B  5  cosh v £ 2  io  B  Y  [  +  ^  cosh Y ft  io  =  (10.10) c  io  D  io  -  A  l l  B  l l  85  cosh Y £ 5 5  Z  5  sinh Y & 5 5 (10.11)  "11  C  D  11  11  f-sinh y £ 5  cosh  5  Y ^ 5  5  We know t h a t t h i s r e p r e s e n t s a s e t o f two t e r m i n a l p a i r networks i n cascade. The  t r a n s f e r m a t r i x o f such a system i s an ordered  individual  product  o f a l l the  matrices 11  TT T.  T =  i=l I t then  (10.12)  1  follows  I  T = T  L  T I  (10.13)  T  J  or 9  TT  (10.14)  T.  i=l  where V , T  1^ a r e the v o l t a g e and c u r r e n t a t the r o o t o f the r e s o n a t o r , V , 1^  are the v o l t a g e and c u r r e n t a t the h o t arm t i p i n the r e s o n a t o r , v o l t a g e and c u r r e n t a t the t r a n s m i s s i o n l i n e (9.17)  and  i s g i v e n by eqn. ( 1 0 . 2 0 ) .  input.  constant.  z  i  a r e the  M' i s g i v e n by eqn.  £^ a r e l e n t g h s o f the f i r s t Dee, o f  the second Dee, and o f the t r a n s m i s s i o n l i n e s e c t i o n s . propagation  V J , I J  The i n p u t impedance i s determined  i s the complex from  (10.15)  =T  Matching the r e s o n a t o r shunt impedance to the tube o p e r a t i n g impedance i s the primary fixed,  t a s k o f the computer code RESLINE. the parameters o f the resonant  Once the tube parameters have been  l i n e and t h e c a p a c i t o r s a r e v a r i e d u n t i l  - 86 the d e s i r e d i n p u t impedance i s a t t a i n e d .  The maximum power t r a n s f e r occurs  when Z,  Besides  UBE  (10.16)  the parameters n e c e s s a r y  to a c h i e v e  a match the program computes a  number of o t h e r parameters at v a r i o u s p o i n t s i n the system.  The program  computes the q u a l i t y f a c t o r s , power l o s s e s , v o l t a g e s , c u r r e n t s , impedances, phases, s t o r e d e n e r g i e s , lumped constant a maximum v o l t a g e a l o n g  the l i n e .  e q u i v a l e n t s , s t a n d i n g wave r a t i o s and  A complete l i s t  o f a l l parameters computed  by the program can be found i n Appendix A. We w i l l now frequency  d e s c r i b e a second method which assumes t h a t the  i s different  from the r e s o n a t o r  a r e s o n a t o r which c o n s i s t s of one Dee p a r a l l e l resonant  c i r c u i t with  impedance measured a c r o s s  resonant  [Fig. 42(a)].  lumped c o n s t a n t s  frequency.  generator  L e t us  Representing  Rp, L, C we f i n d  consider  a Dee as a t h a t the  the t e r m i n a l s DD' i s  (10.17) co  Transforming  03  0  t h i s impedance i n o r d e r  matching network at a c o u p l i n g p o i n t  to o b t a i n the impedance seen by the ( t e r m i n a l s BB') r e s u l t s i n CO  1 03 T~2- 772" k w  2  2  =  0  R  P -  J  R  P  1 + <f  Q  C0  (10.18)  o  co  wo  cog co  where k, the r e s o n a t o r v o l t a g e step-up r a t i o , i s d e f i n e d i n S e c t i o n I I . 9 . 2 . the neighbourhood of f  the r e a c t i v e component  of Z" i s s m a l l .  In  We r e p l a c e Z'  (10.19) with  too  (10.20)  by  - 87 to 2  (10.21)  o In the v i c i n i t y o f f where V  1  the system can be r e p r e s e n t e d  as shown i n F i g . 42(b),  , i s the tube v o l t a g e , L  i s the r e s o n a t o r peak v o l t a g e , V„,  TTT>1:  lUrSrL  loop s e l f i n d u c t a n c e , C  g  i s the c a p a c i t a n c e  transformation r a t i o , V  t h a t p r o v i d e s a proper  i s the loop induced v o l t a g e  In o r d e r to meet the f i r s t  (jtoL I  c o n d i t i o n , i . e . to achieve  T  Li  i s the  voltage  not i n c l u d e d ) . a proper  voltage  t r a n s f o r m a t i o n r a t i o we must have x  V  L  = V  c  +  —  TUBE  X  h  U  We note t h a t f o r | v „ J < | V j the c a p a c i t o r C TUBE L S  (10.22) J  would have to be r e p l a c e d by an  TT1J  , V , X = toL the value, o f IUBE L L L X =-1/(toC ) can be determined and consequently C can be f i x e d . We are now u s s inductance.  Now  f o r given values of V  l e f t w i t h the o n l y unknown  t h a t i s found  from the second c o n d i t i o n  ( r e s i s t i v e i n p u t impedance)  [Re  (Z')]  2  +  [ \ + Imag ( Z ' ) ] [ X  C  + ^  + Imag ( Z ' ) ] =  The phase s h i f t between the r e s o n a t o r v o l t a g e i s obtained and we  as f o l l o w s .  0  (10.23)  and the tube v o l t a g e  A p p l y i n g Thevenin's theorem we s i m p l i f y our c i r c u i t  find -1 F 6 = tan -Ir X  (10.24)  where  To enable  a t r a n s f e r of power from the tube to the Dees a resonant  be used.  The l e n g t h o f the l i n e must be a m u l t i p l e o f h a l f wavelength.  complete c i r c u i t  l o o k s as drawn i n F i g . 4 2 ( b ) .  line  could The  88 10.2  -  Other Resonances of the System Normally a s e c t i o n of t r a n s m i s s i o n l i n e , loaded  a frequency  g i v e n by i t s p h y s i c a l l e n g t h and  l y at a l l i n t e g e r m u l t i p l e s o f t h i s t h a t t h i s might not h o l d i f we along the l i n e .  The  primary  or unloaded, r e s o n a t e s  a f o r e s h o r t e n i n g and  fundamental frequency.  approximate-  I t was  suspected  p l a c e d s e v e r a l c a p a c i t o r s at d i f f e r e n t  t a s k of the computer code MATCH was  at  points  to  i n v e s t i g a t e whether the TRIUMF r e s o n a t o r - t r a n s m i s s i o n l i n e system c o u l d resonate fixed  at f r e q u e n c i e s d i f f e r e n t  the n e c e s s a r y  parameters as two  parameters.  from f The  p a r a l l e l resonant  two  The  the t r a n s m i s s i o n l i n e i s s a i d  the i n p u t impedance i s e q u a l to z e r o . F i g . 43(a)]  Dees were r e p r e s e n t e d w i t h  circuits.  r e p r e s e n t e d w i t h d i s t r i b u t e d parameters. r e s o n a t o r and  f o r which the main program RESLINE  i s the f i r s t  t h i s o p e r a t i o n by  The  resonant  to be  t  k' 7  B ' 7  •  the  i n resonance i f the phase of  F i n d i n g the i n p u t impedance Once a g a i n we  the f o l l o w i n g s e t o f m a t r i c e s : T^, •  was  system c o n s i s t i n g of  o p e r a t i o n of the program.  eqns. (10.1),  line  •  y  lumped  T^,  ....  [see describe  , T^.  T^,  T^,  (10.6). jto(L Lg 2  M  M  T' =  (10.25) -j 7  D  C  1  8  8  2  M  7  B' A  L  0  T" =  (10.26) 1 C  8o  D  8O  R  J  1  2  >  A'  9  B'  9  1  0  T" =  (10.27)  9  D'  9  JwC  2  1  - 89 -  io  B  10  (10.28)  10 D;  10  10  A ' 11  B ' 11  (10.29)  "11 11  11  12  12  jcoCj  1  (10.30)  12  12  12  13  13  1 R.  1 (10.31)  '13 D'  13  The t r a n s f e r m a t r i x  1  13  i s g i v e n by a product 13  Ul The mutual i n d u c t a n c e  (10.32) 1  M i s g i v e n by eqn. (9.13).  V  I  T L  = T" T  l  (10.33)  I  I  «  We can w r i t e  L  >  l  The i n p u t impedance Z^. i s g i v e n by eqn.  (10.15).  Note t h a t i n the v i c i n i t y of each of the resonant resonator, d i f f e r e n t  f r e q u e n c i e s of the  lumped parameters o f the r e s o n a t o r must be used.  program, the impedance seen at the end o f the l i n e i s o b t a i n e d by  In the  transforming  the r e s o n a t o r impedance u s i n g e i t h e r a v o l t a g e step-down r a t i o or a mutual  - 90 inductance.  Although k was d e f i n e d d i f f e r e n t l y than M, the d i f f e r e n c e i n  t r a n s f o r m e d impedances u s i n g e i t h e r k o r M i s v a n i s h i n g l y s m a l l i n t h e frequency range o f i n t e r e s t .  The f o l l o w i n g c o n c l u s i o n s were drawn from t h e  r e s u l t s o f t h e program MATCH. i)  The system  c o n s i s t i n g o f e i t h e r one Dee o r two Dees and a resonant  t r a n s m i s s i o n l i n e has two o r f o u r resonant Other computed resonances ii)  are a s s o c i a t e d with a coupling loop,  One o f t h e computed resonant f r e q u e n c i e s o f t h e system the frequency f yield  a t which the l i n e parameters  a d e s i r e d i n p u t impedance.  The system  maximum power t r a n s f e r o n l y a t t h i s iii)  frequencies, respectively.  i s equal to  were a d j u s t e d t o i s matched f o r a  frequency,  Around f , the system behaves l i k e a simple p a r a l l e l resonant Q  circuit.  The whole system must, t h e r e f o r e , be tuned a t the resonant frequency f of the r e s o n a t o r .  The i n p u t impedance seen by the tube i s p l o t t e d i n F i g . 39.  The  computed resonant f r e q u e n c i e s o f t h e system  The  resonant p r o p e r t i e s a l t e r n a t e between those o f a s e r i e s resonant  and those o f a p a r a l l e l resonant  a r e p r e s e n t e d i n T a b l e XX. circuit  circuit.  The program MATCH a l s o computes v o l t a g e s , c u r r e n t s , power l o s s e s , f a c t o r , impedances and phases i n t h e l i n e . non-resonant Three  quality  The program can accept e i t h e r a  o r i n d u c t i v e l y c o u p l e d and c a p a c i t i v e l y coupled resonant l o a d .  c a p a c i t o r s may be connected  anywhere along the l i n e .  - 91 10.3  P o s s i b l e Operating  Conditions  I f the RF system i s e x c i t e d at a frequency frequency Only  the two  The  are the r e s o n a t o r t i p t o loop v o l t a g e r a t i o and  Once the r e s o n a t o r resonant  frequency  i s known the l i n e s h o u l d be tuned  at  this  run at a frequency which i s s l i g h t l y  from the r e s o n a t o r resonant  r e s o n a t o r resonant The  as i n d i c a t e d by the program.  to o b t a i n the c o r r e c t i n p u t impedance.  However,the system can a l s o be different  the  c u r r e n t d i s t r i b u t i o n s i n the l i n e c l o s e to the computed v a l u e s .  v o l t a g e node p o s i t i o n s h o u l d then be found  frequency  resonant  Dees p r e s e n t a pure r e s i s t i v e l o a d at the c o u p l i n g p o i n t .  f o r t h i s frequency  v o l t a g e and  equal to the r e s o n a t o r  expressions  frequency we  f o r X^,  R^  frequency.  In the v i c i n i t y of  can r e p r e s e n t the system as shown i n F i g . 4 2 ( c ) .  f o r a two  Dee  r e s o n a t o r are c o m p l i c a t e d .  f o r a one  Dee  r e s o n a t o r they are g i v e n by eqns. (10.20) and  which one  f i n d s the range o f f r e q u e n c i e s f o r which the l i n e w i t h  c a p a c i t o r s , loop r e a c t a n c e and which transforms  R^  the r e s o n a t o r r e a c t a n c e  into R^^.  I f the l i n e parameters  were f i x e d , t h e r e would be j u s t one  frequency  d e s i r e d i n p u t r e s i s t a n c e ) c o u l d be o b t a i n e d . and because jwL^  the  and j X ^ are frequency  (10.21)  However, from  connected  c r e a t e s a IT network (capacitors included)  f o r which a proper match ( a S i n c e our c a p a c i t o r s are v a r i a b l e  dependent, the c o n d i t i o n to be  satisfied  to produce a r e s i s t i v e i n p u t impedance i s  [Re  (Z-)]  2  +  [\  + \  i  m  + Imag ( Z ' ) ] ^  Under these c o n d i t i o n s the t r a n s m i s s i o n l i n e and possess  the same resonant  V /VJ*, may  frequency.  be e i t h e r h i g h e r or lower  Consequently,  The  + X  c  + Imag  the r e s o n a t o r no  depending on the o p e r a t i n g  (Z')]=0  longer  r e s o n a t o r t i p to loop v o l t a g e  i t i s p r i m a r i l y the v o l t a g e and  t h a t are a f f e c t e d .  + X^^  ratio,  frequency.  c u r r e n t d i s t r i b u t i o n s i n the  For lower v a l u e s of t h i s r a t i o the v o l t a g e i n the l i n e  e a s i l y be h i g h e r by a f a c t o r of two  than the computed v a l u e s .  In t h i s  line may  case,  - 92  -  the whole system i s i n resonance but at the frequency be  f e d i n t o the r e s o n a t o r but  considerably increase.  The  f ^ f .  Power can  the l o s s e s i n and near the c o u p l i n g loop  still  may  tube w i l l , however, c o n t i n u e to f e e d the same  power i n t o the system as l o n g as the tube sees  the same r e s i s t i v e  input  impedance. For many reasons  such as temperature t r a n s i e n t s i n the r e s o n a t o r  c o o l i n g water p r e s s u r e v a r i a t i o n s , m u l t i p a c t o r i n g e t c . , i t i s not that the RF system w i l l be operated The  frequency  s h i f t s due  at a f i x e d frequency  a t the  panels,  anticipated  beginning.  to temperature t r a n s i e n t s are too l a r g e to be  compensated f o r by means of t u n i n g b e l l o w s .  The  r e s o n a t o r unloaded  frequency  can be measured p r i o r to h i g h v o l t a g e t e s t s u s i n g a l o o s e c a p a c i t i v e c o u p l i n g . The  l i n e i s then connected  Dees and  a resonant  to the r e s o n a t o r and the system c o n s i s t i n g of  l i n e w i l l be tuned under c o l d c o n d i t i o n s i n o r d e r to  a t t a i n a d e s i r e d r e s i s t i v e impedance at the i n p u t of l i n e a t the resonant  frequency.  The  a c t u a l t u n i n g w i l l be accomplished  resonator  by means of  c a p a c i t o r s i n s e r t e d at t h r e e d i f f e r e n t p l a c e s along the l i n e .  The  loop  taken as a r e f e r e n c e to make sure t h a t the t u n i n g i s b e i n g  done a t the r e s o n a t o r frequency. v o l t a g e r a t i o and  F i g s . 44,  45 show the r e s o n a t o r t i p t o loop  the phase between the r e s o n a t o r and loop c u r r e n t s as  f u n c t i o n s o f d r i v i n g frequency versa).  three  resonator  t i p to loop v o l t a g e r a t i o or the phase s h i f t between the r e s o n a t o r and c u r r e n t s can be  two  f o r a f i x e d r e s o n a t o r frequency  Once the t u n i n g of the system has been f i n i s h e d  d e l i v e r e d to the Dees.  the RF power can  ( p u l s e s of about 1 msec width  and up  T h i s p u l s i n g of the a m p l i f i e r w i l l be stopped  r e s o n a t o r v o l t a g e has  vice be  In o r d e r to overcome m u l t i p a c t o r i n g , at the s t a r t ,  main RF a m p l i f i e r w i l l be p u l s e d 140 kV a m p l i t u d e ) .  (and  reached  o s c i l l a t o r y mode o c c u r s r e s o n a t o r frequency w i l l  about 20 kV,  the  to  once the  a t which moment a s w i t c h to a  ( d r i v i n g s i g n a l s u p p l i e d by the r e s o n a t o r ) .  self-  The  change owing to temperature t r a n s i e n t s , c o o l i n g water  p r e s s u r e v a r i a t i o n s and the RF f o r c e s .  The  o p e r a t i n g frequency w i l l  d r i f t , because, f o r a s e l f - o s c i l l a t o r y system, the phase s h i f t feedback loop i s always m a i n t a i n e d o p e r a t i n g frequency which i s now frequency  the phase s h i f t  o p p o s i t e phase s h i f t  e q u a l to 360  around the whole  (or 0 deg).  s l i g h t l y d i f f e r e n t from the  At t h i s  components.  i n the r e s o n a t o r frequency  as l a r g e as -30  kHz  and the o p e r a t i n g  frequency  d i f f e r e n c e i s small  would not a f f e c t  kHz  (CRM).  T h i s frequency  the v o l t a g e d i s t r i b u t i o n i n the f i r s t  s e c t i o n of  the l i n e i n o r d e r to o b t a i n a r e s i s t i v e i n p u t impedance was  Should  t u n i n g of  done a t the  the r e s i s t i v e i n p u t impedance be a t t a i n e d by t u n i n g the l i n e a t  r e s o n a t o r frequency  d i f f e r e n t by more than Af = f - f  the v o l t a g e d i s t r i b u t i o n i n the f i r s t  would be s i g n i f i c a n t l y changed. r e s o n a t o r frequency would now and,  consequently,  e i t h e r go up or down. frequency  and  .  the frequency which was  Af = f - f  would  line  (compared to computed v a l u e s f o r f = f ) p r o v i d e d t h a t the i n i t i a l  resonator frequency.f  an  Computer r e s u l t s  l e a d to a d i f f e r e n c e between the r e s o n a t o r frequency of a t most -0.5  new  resonator  i n t r o d u c e d by the r e s o n a t o r i s compensated by  i n t r o d u c e d by the other RF  have shown t h a t the d r i f t  deg  then  can be  = ±1 kHz  s e c t i o n of the  In a d d i t i o n to t h i s , the d r i f t  l e a d to e i t h e r a decrease  f a r from the r e s o n a t o r frequency  i n the  s e c t i o n would  t h a t the d r i f t  c o n s i d e r e d as a second o r d e r e f f e c t and may  o n l y i f the t u n i n g of the t r a n s m i s s i o n l i n e was  line  o r an i n c r e a s e o f  the v o l t a g e l e v e l i n the f i r s t  I t should be understood  from the  i n the be o f  resonator  significance  c a r r i e d out at a frequency  very  o r i f the RF system i s run i n a f i x e d  frequency mode. Once the d r i f t o s c i l l a t o r y mode drifts  i n the r e s o n a t o r frequency  to the f i x e d frequency  i n r e s o n a t o r frequency  o s c i l l a t i o n s ) d u r i n g the f i x e d  ceases, a s w i t c h from the  mode can take p l a c e .  (due to a r e a c t i v e beam l o a d and  Any  self-  sudden  mechanical  frequency mode w i l l be c o r r e c t e d by means of  - 94 tuning bellows.  Uncompensated r e s o n a t o r  i n c r e a s e i n the power i n p u t i n o r d e r given l e v e l . resonator and  frequency  s h i f t s would r e q u i r e an  to keep the r e s o n a t o r  peak v o l t a g e a t a  Both the power i n p u t i n c r e a s e and the d i f f e r e n c e between the  frequency  and the o p e r a t i n g  c u r r e n t l e v e l s i n the f i r s t  frequency  c o u l d i n f l u e n c e the v o l t a g e  s e c t i o n o f the l i n e .  - 95 -  CHAPTER I I I .  EXPERIMENTAL TESTS  To reduce the c o s t and space requirements done at h a l f - s c a l e . l e v e l s with  most model measurements were  The t e s t s were c a r r i e d out both a t medium and low power  combinations  c o n s i s t i n g of v a r i o u s numbers of resonant s e c t i o n s .  By a medium power l e v e l one means t h a t v o l t a g e s o f about 500 V were reached c a v i t y w h i l e a low power l e v e l r e p r e s e n t s v o l t a g e s of a few V o l t s . segments were made up of copper covered plywood. 're employed to exclude  on  The  Insulators (polyesterene)  we-  h o l d the i n d i v i d u a l p a n e l s i n a c o r r e c t p o s i t i o n and to  any mechanical  oscillations.  To ensure good c o n t a c t s everywhere the  segments were b o l t e d a t the r o o t i n both the h o r i z o n t a l and v e r t i c a l directions.  A l l segments were a l s o h e l d t o g e t h e r r i g i d l y near the t i p .  S i n c e a l l dimensions were s c a l e d down by a f a c t o r o f two the r e s o n a t o r c h a r a c t e r i s t i c impedance remained c o n s t a n t . i n g c a p a c i t a n c e were reduced frequency,  The l e n g t h and the t i p l o a d -  by a f a c t o r o f two.  This resulted i n a  doubled  h i g h e r power l o s s and lower q u a l i t y f a c t o r compared w i t h a f u l l -  s c a l e r e s o n a t o r , see S e c t i o n s  II.1.6.  The c o n t a c t s between the hot arm t i p s were p r o v i d e d by s e v e r a l cm long copper s t r i p s s o l d e r e d t o the hot arms.  I t was found  t h a t the h o t arm  t i p c o n t a c t s i n f l u e n c e d the e l e c t r i c a l p r o p e r t i e s o f the c a v i t y .  tip-to-  For example,  the q u a l i t y f a c t o r o f the r e s o n a t o r w i t h no c o n t a c t s between the h o t arm dropped by a f a c t o r of f i v e . gaps between the p a n e l s .  T h i s was m a i n l y  I t was a l s o found  due to a f l u x leakage  tips  through the  t h a t the v o l t a g e u n i f o r m i t y along  the a c c e l e r a t i n g gap depended s i g n i f i c a n t l y on the q u a l i t y of t h i s  contact.  The wider the copper c o n n e c t i n g s t r i p s a c r o s s the gaps between the r e s o n a t o r segments the b e t t e r was the v o l t a g e u n i f o r m i t y along the a c c e l e r a t i n g gap d u r i n g the model measurements. The q u a l i t y f a c t o r s were always measured w i t h a v e r y l o o s e c a p a c i t i v e c o u p l i n g between the r e s o n a t o r and the g e n e r a t o r .  The measurements at low and  - 96 medium power l e v e l s showed no d i f f e r e n c e i n the q u a l i t y f a c t o r . the Q's measured on i n d i v i d u a l r e s o n a t o r s a r e p r e s e n t e d The  The v a l u e s of  i n the next  Sections.  computed v a l u e s are i n b r a c k e t s . C a p a c i t i v e probes made up o f s o l i d  s t a t e diodes were employed to measure  r e l a t i v e v o l t a g e s i n the upper r e s o n a t o r segments. p r i o r t o any measurements by means of a t h e r m i o n i c  The probes were diode.  During  v o l t a g e s were r e l a t e d t o a r e f e r e n c e probe v o l t a g e and r e l a t i v e d i f f e r e n c e s were r e c o r d e d . drop by about 5%  calibrated  the t e s t s a l l  voltage  The e f f e c t o f t h e c a p a c i t i v e probes on the Q  ) was s m a l l and, t h e r e f o r e , n e g l e c t e d .  (a  MEASUREMENT OF A.  RESONATOR PARAMETERS  Resonant frequency In general, e s t a b l i s h i n g  p r e s e n t e d no or  two  problems.  sections  impedance per B.  are  However, some i n a c c u r a c y might be  assembled.  u n i t w i d t h of the  Tip loading The  the resonant frequency a c c o r d i n g to eqn.  The  v a l u e of t h i s c a p a c i t y  was  accuracy.  measure the r e s o n a t o r l e n g t h  I f we  now  i n C j . = 7.5 p  mean that  depending on Quality As  p'F  A  per  that  to c a l c u l a t e Z and  the  r e s o n a t o r segment at h a l f - s c a l e . T I  p = 15  pF.  The  For  CRM  frequency  This  (1.5) would CRM  the exact v a l u e tank.  factors quality factor i s d i r e c t l y proportional  e f f e c t of v a r y i n g  the v a l u e of the  i s shown i n F i g . 7.  The  measured q u a l i t y f a c t o r s v a r i e d  c o n t a c t s at the  and  between the hot  root  the Q was  to  the  i n a c c u r a c y i n Z ^ r e f l e c t s i n the v a l u e of the  r e s o n a t o r parameters on  arm  tips.  quality  depending upon  Certainly  the main  Q.  factor the factor  a m e c h a n i c a l misalignment of i n d i v i d u a l segments.  The  segments tuned t o s l i g h t l y d i f f e r e n t resonant f r e q u e n c i e s cause a drop i n o v e r a l l Q.  The  quality factors  that were measured on h a l f - s c a l e  w i t h good e l e c t r i c a l c o n t a c t s and were 4600 (5500) and respectively.  The  6100  fundamental and  q u a l i t y f a c t o r s measured on  model amounted to 6000 (7200) and  the  resonators  m e c h a n i c a l alignment of i n d i v i d u a l segments  (9500) f o r the  9300 (12500).  the  t h i r d harmonic,  a full-scale single  a  w i t h a good  l a t e s t measurements on  the n a t u r e of the beam probe h o u s i n g i n the  the v a l u e f o r the  influencing  q  resonant  the measured v a l u e i s around 13 pF,  c h a r a c t e r i s t i c impedance any The  i t i s possible  (1.5).  s u b s t i t u t i o n o f measured v a l u e s i n eqn.  a f u l l - s c a l e value i s C  resonators indicate  C.  sections  e a s i l y e v a l u a t e C^p*  resulted  characteristic  large.  determined from the e q u a t i o n  of 20  can  the  one  capacity  resonator consisting  we  encountered when  e f f e c t o f f l u x guides on  resonator i s very  (1.5)  section  - 98 D.  Phase measurements The phase t e s t s at h a l f - s c a l e were aimed at f i n d i n g out what  the v o l t a g e  phase d i f f e r e n c e between the two extreme ends of a Dee and a c r o s s the accelerating  gap was. The RF v o l t a g e  made up o f 20 s e c t i o n s was  phase d i f f e r e n c e measured on a r e s o n a t o r  found to be l e s s than 1 deg.  Also  the RF phase  d i f f e r e n c e between the upper and lower row o f r e s o n a t o r segments d i d n o t exceed 1 deg.  The phase d i f f e r e n c e  Dees made up of 10 s e c t i o n s was n e i t h e r However, avoided. only.  a c r o s s t h e a c c e l e r a t i n g gap when the two  each were assembled amounted  a f f e c t e d by a h o t arm d e f l e c t i o n n o r by s h i f t i n g  any leakage o f the RF energy from the a c c e l e r a t i n g  the r o o t  plunger.  gap had to be  The t e s t s were done at a medium power l e v e l w i t h c o u p l i n g  to one Dee  Owing to the s a t i s f a c t o r y r e s u l t s i t was d e c i d e d to e x c i t e the r e s o n a t o r  system by means of a s i n g l e c o u p l i n g E.  to 180 ± 0.5 deg and  Resonator lumped  loop.  capacity  The r e s o n a t o r lumped c a p a c i t y was determined by i n s e r t i n g a c a p a c i t o r o f a known v a l u e at the h i g h v o l t a g e resonant f r e q u e n c y .  end of the r e s o n a t o r and measuring the new  Given the o r i g i n a l resonant frequency f , the new Q  frequency f ^ and the a d d i t i o n a l c a p a c i t y resonator i s calculated  C  =  AC, the lumped c a p a c i t y  resonant  of the  from  M  <w21  The t e s t was done w i t h a two s e c t i o n r e s o n a t o r modelled a t h a l f - s c a l e .  Four  capacitors,  tips  and  f  a p p r o x i m a t e l y 1 pF each, were connected between the h o t arm  the grounding p l a t e .  = 45.7146 MHz, f o  The measured v a l u e s of f , f , AC were: AC = 4.3 pF, Q  = 45.3803 MHz, Z  1  = 36.4 ft, I = 1.5619 m, C -  o  A f t e r s u b s t i t u t i o n above the v a l u e of C = 291 pF was o b t a i n e d . t h e o r e t i c a l c a l c u l a t i o n of the lumped c a p a c i t y measured v a l u e was w i t h i n  = 7 pF.  TIP The  r e s u l t e d i n C = 302 pF.  5% of the computed one.  r  The  - 99 -  2.  FREQUENCY TUNING 2.1  Tuning  Stub  I t i s w e l l known t h a t the r e s u l t a n t f r e q u e n c i e s o f two coupled  circuits  can be i n f l u e n c e d by v a r y i n g the parameters o f b o t h c i r c u i t s s i m u l t a n e o u s l y or by changing  the parameters o f one c i r c u i t o n l y .  we have two resonant  c a v i t i e s coupled  The same d e d u c t i o n s h o l d i f  together.  L e t us assume t h a t i n p l a c e o f the o t h e r r e s o n a t o r we i n t r o d u c e a resonant l i n e , approximately and  a q u a r t e r wavelength l o n g , s h o r t c i r c u i t e d a t the f a r end  coupled e i t h e r c a p a c i t i v e l y o r i n d u c t i v e l y to the r e s o n a t o r .  way t o tune the system c o n s i s t i n g o f the r e s o n a t o r and a resonant i s t o v a r y the p o s i t i o n o f the t u n i n g stub p l u n g e r .  The s i m p l e s t t u n i n g stub  Any such change r e p r e s e n t s  a change i n both a lumped c a p a c i t y and i n d u c t a n c e o f the t u n i n g stub Consequently  the resonant  f r e q u e n c y o f the system i s a l t e r e d .  ( F i g . 59).  I n our system  the magnitude o f the frequency v a r i a t i o n a l s o depends on the s e l e c t e d c o u p l i n g point. For the t e s t s s e v e r a l r e s o n a t o r s were modelled  at h a l f - s c a l e .  The main  aim of these t e s t s was to i)  i n v e s t i g a t e the range o f the r e s o n a t o r t u n i n g by means o f t u n i n g stubs and  ii)  f i n d a v o l t a g e v a r i a t i o n a l o n g the a c c e l e r a t i n g gap caused by s h i f t i n g a  The stub  t u n i n g stub  plunger.  t e s t s a l s o i n c l u d e d a measurement o f the q u a l i t y f a c t o r s o f the c a v i t y system. The  r e s u l t s o b t a i n e d w i t h a t e n s e c t i o n model a r e p l o t t e d i n F i g s . 46,  Four stubs coupled c a p a c i t i v e l y were connected #3 and #8 (two p e r upper row, two p e r lower  row of r e s o n a t o r segments).  R c h a r a c t e r i s t i c impedances o f the system were Z o J  magnitude o f t h e c o u p l i n g c a p a c i t a n c e was C  to the r e s o n a t o r i n s e c t i o n s  ST o  = 4 ft,Z =43.6ft.  = 18 pF.  The  The v o l t a g e i n a  The  47.  - 100 p a r t i c u l a r s e c t i o n was s u b j e c t t o two e f f e c t s .  First  the resonant  of t h i s system was dependent on the stub s h o r t i n g p l u n g e r due  frequency  position.  t o t h e f a c t t h a t the c o u p l i n g was n o t u n i f o r m l y d i s t r i b u t e d  Secondly  along t h e  a c c e l e r a t i n g gap t h e f i e l d i n the r e s o n a t o r i n the p r o x i m i t y o f the t u n i n g stub was d i s t o r t e d .  C a p a c i t i v e pick-up  probes made up o f s o l i d  state  diodes  measured v o l t a g e s w i t h r e s p e c t t o a r e f e r e n c e probe v o l t a g e i n s e c t i o n #4. A l l probes were c a l i b r a t e d p r i o r t o measurements. frequency The  and q u a l i t y f a c t o r v a r i a t i o n s measured on a f i v e s e c t i o n r e s o n a t o r .  stub was coupled  R Z  q  Graphs i n F i g . 46 show t h e  c a p a c i t i v e l y i n s e c t i o n #3.  The parameters were:  ST = 4.4 ft, Z  variation  q  = 43.6 ft, C^ = 10.2 pF.  F i g . 46 a l s o shows a  frequency  measured on a twenty s e c t i o n model (each Dee made up o f 10 s e c t i o n s ) .  Two stubs coupled and bottom.  c a p a c i t i v e l y were connected i n each Dee i n s e c t i o n #9, top  A corresponding  v o l t a g e v a r i a t i o n i s shown i n F i g . 47. The R ST  f o l l o w i n g parameters d e s c r i b e t h e system: Z  q  = 4 ft, Z  q  = 43.6 ft, C^ = 11 pF.  In a l l t e s t s mentioned, t h e c o u p l i n g p o i n t was near t h e h i g h v o l t a g e end of the r e s o n a t o r .  The t e r m i n a t i n g c a p a c i t a n c e C^, was produced by t h e p l a t e - t o -  hot arm c a p a c i t a n c e .  I t was deduced that the v o l t a g e v a r i a t i o n c o u l d n o t be  h e l d w i t h i n the r e q u i r e d t o l e r a n c e s and moreover, the q u a l i t y f a c t o r showed a s i g n i f i c a n t drop.  I n the case o f a p u s h - p u l l mode i n t h e r e s o n a t o r - s t u b  a v e r y h i g h v o l t a g e would develop 2.2  system  at a coupling point.  Ground Arm D e f l e c t i o n T h i s k i n d o f t u n i n g was i n v e s t i g a t e d on a f i v e s e c t i o n model assembled a t  half-scale.  The hinged  ends  [see F i g . 15(b)] of both  the upper and lower  ground arms, i . e . t h e p l a t e s o f r e a l dimensions 12 i n . x 16 i n . , c o u l d be moved i n so t h a t a d e f l e c t i o n up t o Ag = - 1.5 cm at t h e t i p was produced. max frequency conclude  The  v a r i a t i o n due to the d e f l e c t e d ground arm t i p was q u i t e l a r g e and we t h a t t h i s method o f t u n i n g c o u l d be used both f o r f i n e and c o a r s e  - 101 frequency  tuning.  The measured v a l u e s , p l o t t e d i n F i g . 1 8 ( a ) , were i n good  agreement w i t h the computed ones. v a l u e s f o r l a r g e r d e f l e c t i o n s may grounding  -  D i s c r e p a n c i e s between computed and measured be accounted  p l a t e c a p a c i t a n c e which was  f o r a change i n the hot  not i n c l u d e d i n c a l c u l a t i o n .  f a c t o r s were u n a f f e c t e d by the d e f l e c t i o n of the ground arm Fig. 18(b)].  The  the resonant frequency  The  s e c t i o n s were a l i g n e d p r o p e r l y and  tuned  [see If a l l  thus t o the same  the measured v o l t a g e s i n d i f f e r e n t s e c t i o n s should always g i v e the tips.  v o l t a g e i n our case i n d i c a t e t h a t the m e c h a n i c a l segments was  D i f f e r e n t v a l u e s of  alignment  of  individual  not p e r f e c t .  Cylindrical  Capacitors  E i g h t c y l i n d r i c a l c a p a c i t o r s [see F i g . 1 5 ( c ) ] , 4 i n . i n diameter, installed #4,  #5,  quality  c o r r e s p o n d i n g v o l t a g e v a r i a t i o n i s shown i n F i g . 49.  same v a l u e f o r any d e f l e c t i o n of the ground arm  2.3  tips  arm-to-  i n a ten s e c t i o n r e s o n a t o r i n both upper and  #6,  #7.  r e s o n a t o r segments  S i n c e the a d d i t i o n a l c a p a c i t a n c e produced by moving the copper  c y l i n d e r s inwards was near the gap  lower  were  was  not spread u n i f o r m l y along the a c c e l e r a t i n g gap  d i s t o r t e d to a c e r t a i n e x t e n t .  I t was  the  a l s o expected  field  that a  drop i n q u a l i t y f a c t o r s would occur as a r e s u l t of d i s t o r t e d e q u i p o t e n t i a l s . The  frequency  v a r i a t i o n i s shown i n F i g . 5 0 ( a ) , w h i l e F i g . 50(b)  q u a l i t y f a c t o r s were a f f e c t e d by  changing  was  C  TIP "  7  P F  '  (see F i g . 51).  t h a t t h i s k i n d of t u n i n g c o u l d not be employed.  c h a r a c t e r i z e d by  the  the p o s i t i o n of these c a p a c i t o r s .  The v o l t a g e v a r i a t i o n exceeded the p e r m i s s i b l e l i m i t s c o n c l u s i o n was  shows how  the f o l l o w i n g parameters: Z  q  The  = 43.2 ft, £ = 1.55  The resonator m,  - 102 2.4  -  Capacitive Plates An  i n v e s t i g a t i o n was  half-scale.  c a r r i e d out on a t e n s e c t i o n model c o n s t r u c t e d  C a p a c i t i v e p l a t e s made up m,  from copper covered  dimensions 4 m x 0.11  m x 0.01  manner t h a t p e r m i t t e d  the t i p l o a d i n g c a p a c i t y to be  were mounted on the ground arm  moving the p l a t e s down towards the hot the p l a t e s and resonator  plywood, of  the ground arms was  arm  tips.  achieved  The  increased  [see F i g . 1 5 ( a ) ]  a c t u a l c o n t a c t between  by means of copper s t r i p s , 4 per  the ground arms should have been made.  system are p r e s e n t e d  i n Section III.2.3.  shown i n F i g . 16, which shows r e a s o n a b l e  The  The  a  not  measured and  r e l a t e d to a r e f e r e n c e probe i n s e c t i o n #2  2.5  the  change i s  the computed  Once a g a i n the e f f e c t of c a p a c i t i v e p l a t e s on the t i p - t o - g r o u n d e d c a p a c i t y was  continuous  parameters of  measured frequency  agreement w i t h  included i n c a l c u l a t i o n .  real  tips i n a  segment, however, to a v o i d any d i s t o r t i o n of the f i e l d  contact with  at  values.  plate  A percentage v o l t a g e d i f f e r e n c e as i s plotted in Fig.  I n d u c t i v e Loops at the Root S e v e r a l models were assembled at h a l f - s c a l e to i n v e s t i g a t e t u n i n g of  c a v i t y by means of r o t a t i n g loops and f i n s , e v e n t u a l l y , the l o o p s , are not  f i n s as shown i n F i g . 1 5 ( d ) . too f a r a p a r t we  can v a r y  their axis.  The  e f f e c t on the resonant  frequency  due  the  effective respect  to a d i s t o r t i o n  the f i e l d near the r o o t i s e q u i v a l e n t to the e f f e c t caused by s h i f t i n g root  the  I f the  l e n g t h of the r e s o n a t o r by r o t a t i n g e i t h e r the loops or the f i n s w i t h to  52.  of  the  plane. The  frequency  v a r i a t i o n measured on a t h r e e s e c t i o n r e s o n a t o r  s c a l e i s shown i n F i g . 53.  S i x t e e n f i n s , 0.5  i n . x 1.5  i n . x 2.5  at h a l f i n . , were  mounted i n the m i d d l e segments.. A l l the f i n s were g r a d u a l l y t u r n e d  from  h o r i z o n t a l to the v e r t i c a l p o s i t i o n .  and  were r o t a t e d = 4560.  (see F i g . 53). I t can be  The  In the next t r i a l the f i n s #3  q u a l i t y f a c t o r of an unperturbed c a v i t y  the #6 was  seen from the graphs i n F i g . 53 t h a t the q u a l i t y f a c t o r  by  - 103 i s a f f e c t e d by the i n s e r t i o n of f i n s i n s i d e the c a v i t y and f u r t h e r a l t e r e d i f they  are rotated.  T h i s i s due to h i g h l o s s e s i n the f i n s due to the h i g h  magnetic f i e l d d e n s i t y near the r o o t and the consequent huge induced  currents  i n the f i n s . The  second s e t of experiments was done w i t h  a two s e c t i o n r e s o n a t o r  e i t h e r f i n s o r l o o p s were mounted on t o the r o o t p l u n g e r s . f i n s p e r each segment.  There were e i g h t  The q u a l i t y f a c t o r s of the r e s o n a t o r without  elements were 4250 and 5950 f o r the f i r s t  where  tuning  and t h i r d harmonic, r e s p e c t i v e l y .  Both the l o o p s and the f i n s were made up o f the same dimensions, 0.5 i n . x 1.5 i n . x 2.5 i n . .  The loops were made from a 0.02 i n . t h i c k copper.  The  r e s u l t s a r e p l o t t e d i n F i g s . 54, 55.  2.6  T h i r d Harmonic Tuning Diaphragms The measurements done a t h a l f - s c a l e i n c l u d e d p r a c t i c a l t e s t s w i t h  frequencies  i n j e c t e d i n t o the r e s o n a t o r  c o u l d be o b t a i n e d without  two  and were to prove t h a t a "square wave"  great d i f f i c u l t y .  Using  two 50 PJ c o a x i a l  t r a n s m i s s i o n l i n e s and e x c i t i n g b o t h harmonics s e p a r a t e l y by means o f s e p a r a t e c o u p l i n g l o o p s , the r e s o n a t o r  c o u l d be operated  a t v a r i o u s m i x t u r e s of the  fundamental and the t h i r d harmonic where the t r i p l e r was used. need f o r an a d d i t i o n a l t u n i n g element i n o r d e r r a t i o was e n v i s a g e d ,  Though the  to o b t a i n a proper  no t u n i n g diaphragms were used at t h i s time.  system c o n s i s t e d o f s e v e r a l coupled  c i r c u i t s and the r e s u l t a n t  f r e q u e n c i e s f , f ^ were the f r e q u e n c i e s  o f the e n t i r e system.  frequency The whole  resonant By means o f  feedback v i a the t r a n s m i s s i o n l i n e s the s t a t e c o u l d be found when the r a t i o f g / f ^ = 3 and the d e s i r e d amount o f the t h i r d harmonic was p r e s e n t .  Good phase  s t a b i l i t y was a t t a i n e d owing to i n t e r n a l feedback between two power a m p l i f i e r s . When running with  the two Dees, each made up o f 5 s e c t i o n s , i n p u s h - p u l l mode  c o u p l i n g t o one Dee o n l y t h e r e was no d i f f i c u l t y  wave",as w e l l .  i n o b t a i n i n g the "square  The amount of the t h i r d harmonic c o u l d be v a r i e d from 0 to 20%  - 104 o f the fundamental  l e v e l of 500  V.  S e v e r a l problems were not s o l v e d . frequency f ^ was Q  not t h r e e times f  wave form the fundamental than f  .  The  -  The n a t u r a l t h i r d harmonic r e s o n a t o r  .  Q  To a c h i e v e the d e s i r e d c a v i t y v o l t a g e  a m p l i f i e r was  o p e r a t e d at a frequency s l i g h t l y  t h i r d harmonic a m p l i f i e r was  lower  operated at f  03  °1  These problems were s o l v e d w i t h a s i n g l e s e c t i o n f u l l - s c a l e r e s o n a t o r o r i g i n a l l y designed t o v e r i f y the RF system program RESLINE. made i n f u l l - s c a l e , operated at the nominal fundamental  (see S e c t i o n I I I . 5 ) .  i n . x 25 i n . , were chosen  x 2.5  p o s i t i o n 5 X ^ / 2 4 = 108.5 of 5400 and 6100  As and  This resonator,  frequency 22.66 MHz  t u n i n g elements, connected  two  of the  diaphragms, 0.25  t o the ground arms a t the  i n . from the r o o t (see F i g . 56).  The q u a l i t y  factors  were o b t a i n e d , r e s p e c t i v e l y .  The m e c h a n i c a l  c o n s t r u c t i o n of the diaphragms made i t p o s s i b l e t o t u r n  e i t h e r diaphragm by 90 deg, i . e . from the h o r i z o n t a l to the v e r t i c a l The  in.  position.  frequency v a r i a t i o n as measured w i t h the diaphragms turned g r a d u a l l y from  the h o r i z o n t a l to the v e r t i c a l p o s i t i o n was  an o r d e r of magnitude g r e a t e r than 16  the v a l u e s p r e d i c t e d u s i n g the p e r t u r b a t i o n t h e o r y developed by S l a t e r F i g s . 21 and 5 7 ( a ) ] . proved  The  same measurement r e p e a t e d f o r the t h i r d harmonic  t h a t the fundamental  resonant frequency was  i n f l u e n c e d more than  t h i r d harmonic resonant frequency, but the same disagreement was  [see  found as f a r as the percentage  measured maximum detuning was  change was  concerned.  with  the  calculation  However, the  i n good agreement w i t h the v a l u e s  calculated  u s i n g a d i f f e r e n t method of p e r t u r b a t i o n t h e o r y , namely c o n s i d e r i n g a c a p a c i t i v e e f f e c t produced f  /f  by a t u n i n g diaphragm (see F i g . 22).  vs p o s i t i o n o f diaphragms was  the p o s i t i o n where the r a t i o was A s i n g l e o s c i l l a t o r was  p l o t t e d i n F i g . 57(b)  The  and t h i s  ratio indicated  e x a c t l y e q u a l to 3.  used as a s i g n a l s o u r c e , the t h i r d harmonic  frequency b e i n g generated by a t r i p l e r .  F i g . 56(a) shows the arrangement of  - 105 the experiment a t the time when the r e s o n a t o r was r u n w i t h the two harmonics. Both harmonics were i n j e c t e d at about the same d i s t a n c e from the r o o t , the sensor b e i n g a c a p a c i t i v e probe a t the f r o n t . varied  The t h i r d harmonic amplitude  from 0 to 20% a t the 50 V l e v e l o f the fundamental.  Quality  factors  remained n e a r l y c o n s t a n t d u r i n g motion o f t h e diaphragms. At t h i s s t a g e the fundamental frequency was changed by +3.2% the r o o t p l a n e ) and the r e s o n a t o r was r e t u n e d so t h a t f  /f  (shifting  was a g a i n e q u a l  °3 °i to 3. both  The diaphragms were turned by about 20 deg to r e t u n e the r e s o n a t o r t o frequencies.  As the measured f r e q u e n c y v a r i a t i o n d i d n o t agree w i t h the r e s u l t s based on the p e r t u r b a t i o n t h e o r y which c o n s i d e r e d a change i n the e l e c t r i c and magnetic e n e r g i e s s t o r e d i n the r e s o n a t o r i t was concluded t h a t t o get b e t t e r u n d e r s t a n d i n g of what was happening the p o s i t i o n o f the diaphragms  s h o u l d be  altered. The two diaphragms were now mounted i n the middle o f r e s o n a t o r , X /8 from the r o o t . The r e s o n a n t frequency v a r i a t i o n and the r a t i o f / f oi 03' may be seen i n F i g . 58.  The r a t i o never exceeded 3 s t a y i n g c o n s t a n t over a  range o f diaphragm motion. X  01  01  A f u r t h e r change i n p o s i t i o n o f the diaphragms t o  I l k from the r o o t and s i m i l a r i n v e s t i g a t i o n s as b e f o r e r e s u l t e d i n curves  shown i n F i g . 58.  The r e s u l t s were u n s a t i s f a c t o r y s i n c e the r a t i o , a g a i n , was  always below 3. I t was concluded t h a t the p e r t u r b a t i o n was too b i g , and t h a t a c a p a c i t i v e e f f e c t predominated, the new resonant frequency b e i n g m a i n l y determined by a change i n the t o t a l lumped c a p a c i t a n c e .  T a b l e X I I I summarizes  frequency change produced by moving t h e diaphragms vertical position.  the maximum  from the h o r i z o n t a l t o the  The measured v a l u e s a r e compared w i t h computed ones.  The  low f r e q u e n c y c a p a c i t a n c e i n c r e a s e A C ^ produced by t u r n i n g the diaphragm  from  the h o r i z o n t a l t o the v e r t i c a l p o s i t i o n was measured and found to be e q u a l to 6 pF.  - 106 -  TABLE X I I I Computed  and measured frequency s h i f t s  diaphragms from the h o r i z o n t a l Position  Af  caused by t u r n i n g  to the v e r t i c a l  Af  Af  the  position Af  oi measured (%)  oi computed (%)  measured (%)  computed  275  -  1.73  -  1.87  -  0.91  -  0.99  166  -  0.98  -  1.02  -  0.91  -  0.98  55  -  0.14  -  0.14  -  1.24  -  1.04  from r o o t (cm)  03  03  (%)  - 107  3.  VOLTAGE AND  FREQUENCY VARIATIONS DUE  S i n c e each Dee  TO MECHANICAL MISALIGNMENTS  c o n s i s t s of 40 r e s o n a t o r segments q u i t e s e v e r e  t o l e r a n c e s must be met A misalignment  -  i n o r d e r to a c h i e v e good e l e c t r i c a l  mechanical  properties.  o f e i t h e r r e s o n a t o r p a n e l s or r o o t p l u n g e r s r e s u l t s i n a  d i f f e r e n t c h a r a c t e r i s t i c impedance, r e s o n a t o r l e n g t h and a t i p l o a d i n g capacity.  T h i s l e a d s to a non-uniform  a c c e l e r a t i n g gap.  The  v o l t a g e d i s t r i b u t i o n along the  expected mechanical misalignment  Ail = ±0.040 i n . , i n r e s o n a t o r p a n e l s n e c e s s a r y t o v e r i f y on a model how  i n root plungers i s  ( v e r t i c a l l y ) Ag = ±0.020 i n .  l a r g e the v o l t a g e and  frequency  were i f one of the above mentioned m e c h a n i c a l misalignment The measurements were f i r s t where each Dee was  done on a two Dee  In the f i r s t  variations  occured.  resonator at h a l f - s c a l e  made up o f 10 s e c t i o n s and l a t e r on a one Dee  c o n s i s t i n g of 18 s e c t i o n s .  I t was  graph, F i g . 60, the  resonator  percentage  v o l t a g e change a l o n g the a c c e l e r a t i n g gap vs a d e f l e c t i o n o f the upper h o t in  s e c t i o n #3 was  r e l a t e d to a r e f e r e n c e probe i n s e c t i o n //9.  frequency v a r i a t i o n i s shown i n F i g . 61. v o l t a g e v a r i a t i o n produced measured q u a l i t y f a c t o r was  arm  A corresponding  The next graph, F i g . 62, shows the  by moving r o o t p l u n g e r s i n s e c t i o n #7. Q^ = 3900 w i t h o u t  the probes, Q-^  =  The  3500 w i t h  the  probes i n . S i m i l a r r e s u l t s were o b t a i n e d w i t h a two measurement gave Q one Dee  caused  position.  = 3200.  Dee  r e s o n a t o r , F i g . 6 1 , 62.  I t s h o u l d be noted here t h a t any misalignment  the v i r t u a l n o d a l p l a n e i n the middle  to s h i f t  An asymmetry i n the Dee v o l t a g e s immediately  misalignment  The  w i t h r e g a r d t o r o o t p l u n g e r s caused  v a l u e o f the t h i r d harmonic q u a l i t y  factor  in  from i t s c e n t r a l  followed.  Also  any  a s i g n i f i c a n t decrease i n the  (see F i g . 64).  4.  108  -  RESONATOR MODIFICATIONS 4.1  Extreme End  Segments  A f i v e s e c t i o n r e s o n a t o r was measured and  found to be  t h i r d harmonic, r e s p e c t i v e l y .  one  and  measuring the Q's  = 3000.  The  voltage  assembled and  4600 (5200) and  the  Q^  first  5500 (9080) f o r the  R e p l a c i n g the  a g a i n r e s u l t e d i n the  uniformity  the q u a l i t y f a c t o r s were  fifth low  fundamental  and  s e c t i o n with a tapered  values:  Q^  a l o n g the a c c e l e r a t i n g gap  = 3000, was  also  unsatisfactory. The  extreme end  s e c t i o n was  now  m o d i f i e d i n o r d e r to o b t a i n  c h a r a c t e r i s t i c impedance throughout the modification Fig.  23,  voltage  both the  however, the the  an adjustment of the hot  ground arms and  v a r i a t i o n along the  fundamental.  that  included  The  section  hot  arm  accelerating  q u a l i t y f a c t o r f o r the  (see S e c t i o n  i n power l o s s due  II.5.1).  This  arms i n the extreme end  section,  t i p s b e i n g l e f t unchanged. gap  was  The  r e a s o n a b l y good f o r  fundamental was  t h i r d harmonic Q remained at about Q^  increase  a constant  the  improved to Q  = 4200,  T h i s would  indicate  = 3100.  to a m o d i f i e d extreme end  section  r e p r e s e n t e d a s i g n i f i c a n t f r a c t i o n of the t o t a l amount of the power l o s s i n t h i r d harmonic mode.  The  percentage i n c r e a s e  i s much l a r g e r than t h a t f o r the two  current  f u l l Dee  maxima per  made up  of 20  s e c t i o n s be  fundamental has  lowered by  a f a c t o r of two  made to e n l a r g e the vacuum tank and  s e c t i o n at e i t h e r end  4.2  of a  t h i r d harmonic o n l y one.  b o t h f o r the At  has  Should a  assembled, the percentage i n c r e a s e  t h i r d harmonics (compared w i t h a f i v e s e c t i o n model). was  t h i r d harmonic power l o s s  fundamental, because the  s e c t i o n w h i l e the  t o t a l power l o s s would be  i n the  in  first  t h i s stage a  the and  decision  thus accomodate a normal resonant  Dee.  C e n t r a l Resonator Segments The  measurements s t a r t e d on a 20  the  s e c t i o n model w i t h no  centre post  and  w i t h no c u t - o u t .  The  aim was  f a c t o r s of the unmodified = 2100 to any  (9200).  109 -  to o b t a i n r e f e r e n c e v a l u e s f o r the  model.  The measurement r e s u l t e d i n Q  = 4000  The v a l u e of the t h i r d harmonic q u a l i t y f a c t o r was  i m p e r f e c t i o n s i n the r o o t c o n t a c t s and mechanical  r e s o n a t o r segments.  Two  misalignment  (5250),  sensitive of  c e n t r a l s e c t i o n s were then taken out and r e p l a c e d w i t h  s e c t i o n s w i t h m o d i f i e d hot arm installed.  quality  The v a l u e s of Q  tips  ( c u t - o u t ) and  = 4000, Q  the c e n t r e p o s t was  = 460 were then measured.  also  A  s i g n i f i c a n t improvement i n Q  o c c u r r e d as soon as the r o o t p l u n g e r s i n s e c t i o n s 3 the c u t - o u t were moved i n . The v a l u e i n c r e a s e d to Q = 1400. At t h a t 3  with  time i t was  d e c i d e d not to proceed w i t h f u r t h e r i n v e s t i g a t i o n s and  on a 20 s e c t i o n model and any  change i n mechanical  to use r a t h e r a 4 s e c t i o n r e s o n a t o r c o n s t r u c t i o n should be immediately  e l e c t r i c a l p r o p e r t i e s of the r e s o n a t o r .  I t was  felt  demanded by the beam dynamics group. the c e n t r e was  almost  the a c c e l e r a t i n g gap.  1.5% was  3  = 2500.  occurred.  The  tuned  the  to the same  the  to i n c l u d e changes  d e t a i l e d d e s i g n of the r e s o n a t o r i n  f i n a l i z e d f o r an i n j e c t i o n gap  at 36 deg w i t h r e s p e c t to  The measured v a l u e of the t h i r d harmonic q u a l i t y  The v o l t a g e along the t i p i n the middle  f o r the t h i r d harmonic. almost  reflected in  the t e s t i n g p e r i o d the dimensions of the c e n t r e post and  i n j e c t i o n gap p o s i t i o n have been a l t e r e d s e v e r a l times  Q  where  frequency.  During  was  (one Dee)  t h a t by a l t e r i n g  geometry around the c e n t r e post the f o u r s e c t i o n s c o u l d be resonant  improvements  I t was  factor  s e c t i o n v a r i e d by  about  a l s o deduced from the t e s t s t h a t the Q  u n a f f e c t e d u n l e s s some d r a s t i c a l changes i n r e s o n a t o r  geometry  The v o l t a g e v a r i a t i o n of the fundamental along the a c c e l e r a t i n g gap  i n the c e n t r a l s e c t i o n s was has been proved  always l e s s than  .5%  (see T a b l e I ) .  However, i t  t h a t a good v o l t a g e u n i f o r m i t y along the a c c e l e r a t i n g gap  in  the c e n t r a l s e c t i o n s does not n e c e s s a r i l y mean t h a t the q u a l i t y f a c t o r i s good. On  the o t h e r hand a good q u a l i t y f a c t o r  guarantees  a good v o l t a g e u n i f o r m i t y .  - no All  the measurements were r e p e a t e d once the b a s i c d e s i g n o f the c e n t r e  p o s t and the i n j e c t i o n gap was f i n a l i z e d . a c c e l e r a t i n g gap.  T h i s gap now c o i n c i d e s w i t h the  S i n c e the i n j e c t i o n gap i s produced by a s e p a r a t i o n between  the c e n t r e p o s t and the h o t arm t i p s o f one Dee, o n l y 100 kV v o l t a g e peak w i l l be developed assembled.  there.  A two Dee r e s o n a t o r , each Dee made up o f 4 s e c t i o n s , was  F i g . 65 shows the b a s i c geometry i n the c e n t r a l r e g i o n . A l l  m o d i f i c a t i o n s made concerned  the r e s o n a t o r gap (hot arm t i p - t o - g r o u n d arm t i p )  i n the c e n t r a l r e s o n a t o r segments. of p o i n t s ,  the d i s t r i b u t i o n o f e q u i p o t e n t i a l s near the a c c e l e r a t i n g gap b e i n g  then o b t a i n e d through frequency  The v o l t a g e s were measured a t a g i v e n g r i d  use o f the computer program FITVOLT.  The resonant  and the v o l t a g e n o n - u n i f o r m i t y were then taken as a r e f e r e n c e f o r  the next m o d i f i c a t i o n s .  The t h i r d harmonic Q c o u l d be improved t o Q  - 1500.  -  5.  Ill -  RESONANT LINE AS A MATCHING NETWORK The  aim o f the t e s t s was t o v e r i f y t h e computer program RESLINE.  s c a l e s i n g l e resonant  A full-  s e c t i o n , m o d i f i e d i n o r d e r t o meet the supposed  c h a r a c t e r i s t i c impedance of the r e a l r e s o n a t o r i n the CR c y c l o t r o n , was used. T h i s m o d i f i c a t i o n , i . e . a change i n a nominal s i z e o f one r e s o n a t o r p a n e l , was due  to a d i f f e r e n t e f f e c t o f the f l u x guides  see S e c t i o n I I . 1 . 1 .  on the c h a r a c t e r i s t i c  impedance,  Copper sheeted plywood and aluminum ( f o r the t r a n s m i s s i o n  l i n e l i d ) were the o n l y m a t e r i a l s used.  The l e f t hand p a r t of t h e r e s o n a t o r  system, i . e . t h e o t h e r Dee, was s i m u l a t e d by p u t t i n g a grounded p l a t e i n p l a c e where t h e v i r t u a l grounding The  plane  goes.  r e s o n a t o r was e x c i t e d through  loose inductive or c a p a c i t i v e coupling.  A VHF o s c i l l a t o r and a 10 W a m p l i f i e r were employed as sources  of RF energy.  For m o n i t o r i n g purposes and phase measurements, simple low c a p a c i t a n c e probes were used as RF s e n s o r s . used w i t h  these probes.  Hewlett-Packard  sampling  and v e c t o r v o l t m e t e r s were  Two i d e n t i c a l DC probes r e p r e s e n t i n g a minimal l o a d  (< 1 pF, r e s i s t i v e admittance n e g l i g i b l e ) served to measure the v o l t a g e s the t r a n s m i s s i o n l i n e by t o u c h i n g  the i n n e r conductor  at various  along  stations.  The measured o r computed q u a n t i t i e s were l i n e a r l y s c a l e d i n o r d e r to a l l o w making comparisons. During  the measurements an o p p o s i t e flow of energy was o b t a i n e d , i . e .  from the r e s o n a t o r t o the v i r t u a l a m p l i f i e r a t the i n p u t o f l i n e .  The RF  energy was e x t r a c t e d from the r e s o n a t o r by means of the c o u p l i n g l o o p .  Under  these c o n d i t i o n s the t r a n s m i s s i o n l i n e was loaded a t t h e end by an impedance computed by the program RESLINE.  The r e s i s t i v e p a r t , i n f a c t , s i m u l a t e d the  i n t e r n a l r e s i s t a n c e o f the power tube.  I t s h o u l d be emphasized t h a t the  o p p o s i t e f l o w o f energy had t o be used i n order to minimize the l o a d i n g e f f e c t of s o l i d s t a t e diodes on the output v a l u e s o f the q u a l i t y f a c t o r s  from the power a m p l i f i e r .  (loaded and u n l o a d e d ) ,  The c a l c u l a t e d  loop s e l f - i n d u c t a n c e ,  - 112 -  TABLE XIV C a l c u l a t e d and measured dimensions o f the f u l l - s c a l e resonant s e c t i o n and the r e s o n a n t l i n e Calculated  Hot  (cm)  318.60  319.02  62.36  62.36  1.91  1.91  - ground arm gap  10.16  10.16  - grounded p l a t e gap  7.62  7.62  325.62  326.64  82.68  82.68  - ground arm gap  38.10  38.10  length  44.45  44.45  width  6.35  6.35  thickness  2.54  2.54  height  4.45  4.45  20.32  20.32  length width tip  Ground arm  radius  length width  loop  position  Transmission line  Measured  (cm)  arm  Coupling  single  inner  conductor  1.91 x 5.12  1.91 x 4.74  outer  conductor  7.09 x 10.01  7.09 x 10.01  length  total  vertical  part  675.00  675.00  45.09  45.09  _ 113 _  c h a r a c t e r i s t i c impedance, dimensions of the r e s o n a t o r and l i n e are summarized i n T a b l e s XIV,  XV.  In o r d e r to a r r i v e at a d e s i r e d  c h a r a c t e r i s t i c impedance of the t r a n s m i s s i o n l i n e , l e n g t h was  measured and  the t r a n s m i s s i o n  the c a p a c i t a n c e per  unit  the dimensions of the i n n e r conductor were a d j u s t e d  as  required.  TABLE XV C a l c u l a t e d and measured e l e c t r i c a l c h a r a c t e r i s t i c s of the f u l l - s c a l e s i n g l e resonant s e c t i o n and the resonant l i n e Calculated  f  o!  f  o3  (MHz)  22.66  22.41  (MHz)  -  67.02  7200  5970  12450  9300  l  Q  % Z  o  Z  TL  L  L  Measured  (ft)  38.3  -  (ft)  50.0  50.0  (yH)  0.208  -  Throughout the assembly p e r i o d , the q u a l i t y f a c t o r s of the system were measured a t both  the fundamental and  T a b l e XVI).  amount of power absorbed  The  i n t e r n a l r e s i s t a n c e o f the tube was v a r i e d , and resistance.  the t h i r d harmonic f r e q u e n c i e s  at the r e s i s t o r s i m u l a t i n g the  measured.  the v o l t a g e a c r o s s the r e s i s t o r was The  c a p a c i t a n c e connected  RESLINE, and i t was  (see  The v a l u e of t h i s r e s i s t o r  was  measured f o r each v a l u e o f  a t .this p o i n t was  calculated  h e l d constant d u r i n g the measurements.  by  T h i s c a p a c i t o r was  -  114 -  TABLE XVI Q u a l i t y f a c t o r and power absorbed vs tube s i m u l a t i n g r e s i s t a n c e ^TUBE  Q  L  (V)  (mW)  510  1.088  1.447  2040  1000  1.570  1.825  2640  1500  1.821  1.903  3150  2000  1.982  1.927  3150  2400  2.046  1.878  3240  3000  2.137  1.859  3340  3900  2.243  1.855  3600  4300  2.279  1.852  3660  4700  2.310  1.850  3600  10000  2.456  1.751  3780  22000  2.551  1.712  4100  (ft)  used  P  V  to c a n c e l t h e i n d u c t i v e component of t h e i n p u t impedance.  At the  t r a n s m i s s i o n l i n e i n p u t the system can be r e p r e s e n t e d as a c o n s t a n t c u r r e n t g e n e r a t o r f e e d i n g the i n p u t r e s i s t a n c e i n p a r a l l e l w i t h the tube s i m u l a t i n g resistance.  The maximum power output  when the g e n e r a t o r  from a c o n s t a n t c u r r e n t g e n e r a t o r  conductance i s equal t o the l o a d conductance.  q u a l i t y f a c t o r s h o u l d then drop  t o one h a l f o f the unloaded  one.  occurs  The loaded The measured  v o l t a g e s and c o r r e s p o n d i n g l o s s e s at the end o f l i n e a r e p r e s e n t e d i n T a b l e XVI.  F o r c a l c u l a t i o n o f the power absorbed,  i n p u t r e s i s t a n c e was used,  the computed v a l u e o f the  i . e . R^ = 2083 ft. I t was concluded  t h a t the  i n d i c a t e d v a l u e of r e s i s t a n c e g i v i n g a maximum a b s o r p t i o n i s i n r e a s o n a b l e agreement w i t h the computed v a l u e .  - 115  -  An analogous measurement f o r the t h i r d harmonic frequency o n l y a n e g l i g i b l e f r a c t i o n of the RF energy was i n t o the t r a n s m i s s i o n l i n e . matched the r e s o n a t o r at the fundamental The good  f o r the RF  with  t h a t the  t r a n s m i s s i o n l i n e were i n  Both the measured and  was  T  the v o l t a g e and  the c a p a c i t o r s e t t i n g s .  were changed and  ratio V /V  computed  c u r r e n t d i s t r i b u t i o n s i n the The v a l u e s o f c a p a c i t o r s  the computer program c a l c u l a t e d the new  the l i n e and  at the i n p u t of the l i n e  t h a t i t was  The  by a d j u s t i n g the c a p a c i t o r  (see F i g . 67).  Larger  were a l s o examined  changes i n the p o s i t i o n  (see F i g . 68).  I t was  always p o s s i b l e to a t t a i n a d e s i r e d v o l t a g e r a t i o V / V  a d j u s t i n g some parameter i n the l i n e , however, at the expense of d i f f e r e n t v o l t a g e and v o l t a g e and  c u r r e n t d i s t r i b u t i o n s i n the l i n e .  readjusted  according  discrepancy  the loop and  completely  In o r d e r  that  between Computed and measured v a l u e s  a l s o recorded.  measured v a l u e s was  values, be  T h i s accounts f o r the  i n F i g s . 67 and  68.  the loop and  at the r e s o n a t o r hot  arm t i p  A d i f f e r e n c e of about 7.5%  between the computed  and  found.  across  the  a c r o s s the middle c a p a c i t o r must  to the computed v a l u e , a s w e l l .  r a t i o of v o l t a g e s  proved  by  c u r r e n t d i s t r i b u t i o n s along the l i n e f o l l o w the computed  the r a t i o of v o l t a g e s a c r o s s  resonator  CP(2)  voltage  the d e s i r e d tube s i m u l a t i n g impedance.  kept c l o s e to the computed v a l u e  of the m i d d l e c a p a c i t o r CP(3)  was  values  J_»  j~  The  line  fundamental are p l o t t e d i n the graph i n F i g . 66.  d i s t r i b u t i o n along  CP(6)  the resonant  computed v a l u e s .  a l s o i n v e s t i g a t e d how  CP(3)  our e x p e c t a t i o n  resonator  frequency.  l i n e are a f f e c t e d by and  f l o w i n g from the  shown t h a t  shunt r e s i s t a n c e to the tube o p e r a t i n g r e s i s t a n c e only  v o l t a g e s measured along  agreement  We  T h i s confirmed  had  T h i s was  probably  due  parameters from the computed ones and  coupling loop.  The  discrepancy  coupling loop.  The  v o l t a g e phase was  to d e v i a t i o n s of  the  i n a c c u r a t e p o s i t i o n i n g of  the  c o u l d e a s i l y be removed by r e p o s i t i o n i n g the measured and p l o t t e d i n F i g . 69.  The  "  f o l l o w i n g remarks g i v e p o s s i b l e sources  d u r i n g the i)  116  of d i s c r e p a n c i e s found  tests.  Since  the c h a r a c t e r i s t i c impedance of the r e s o n a t o r was  only  c a l c u l a t e d and not d i r e c t l y measured, an e r r o r up to 10% has  to be  A l l computed q u a n t i t i e s which were d e r i v e d from Z  considered.  q  may  be i n e r r o r . ii)  The  90 deg bend o f the t r a n s m i s s i o n l i n e was  iii)  The  components were not s e l e c t e d w i t h  Also a l l leads connecting  taken i n t o account,  special precautions.  components were temperature dependent and inductive.  not  The  some of the r e s i s t o r s were  capacitors contributed  inductively. iv)  A l l computing was and  C.],j.p.  can be  The  based on the t h e o r e t i c a l v a l u e s  of q u a l i t y f a c t o r s  e f f e c t of the Q on the t r a n s m i s s i o n l i n e parameters  found i n S e c t i o n I I . 8 .  - 117 CHAPTER IV. 1.  CENTRE REGION CYCLOTRON  LOW POWER LEVEL 1.1  Quality Factors As the a c t u a l q u a l i t y f a c t o r s o f t h e CR r e s o n a t o r a r e much h i g h e r  those measured on the h a l f - s c a l e r e s o n a t o r s and because o f f r e q u e n t v i b r a t i o n s o f the r e s o n a t o r panels  the q u a l i t y  (Q — 400. f mech  =  than  mechanical  4 Hz) the d e t e r m i n a t i o n of  f a c t o r s by measuring a decay time c o n s t a n t o f the r e s o n a t o r  fields  was p r e f e r r e d . The measured and computed q u a l i t y f a c t o r s a r e compared i n Table  XVII. TABLE XVII Measured and computed q u a l i t y f a c t o r s o f the CR r e s o n a t o r Computed  Q  Q  3  3  Measured  one Dee  7100  6300  one Dee  12400  9000  two Dees  7100  6250  two Dees  12400  6200  The q u a l i t y f a c t o r s f o r one Dee (4 segments) were measured when the Dee was mounted  i n an a u x i l i a r y frame.  were i n s t a l l e d i n the vacuum  A measurement  tank (see S e c t i o n  w i t h two Dees was done when they IV.1.8).  To a c h i e v e a h i g h q u a l i t y f a c t o r the r e s o n a t o r segments had to be a l i g n e d v e r y a c c u r a t e l y and any leakage  o f t h e RF energy had t o be a v o i d e d .  e l e c t r i c a l c o n t a c t s were n e c e s s a r y  Perfect  a t the r o o t o f the r e s o n a t o r , both between  the r o o t p l u n g e r and the arms o f one r e s o n a t o r segment, between the i n d i v i d u a l r o o t p l u n g e r s , and between the t i p s o f the r e s o n a t o r segments.  Moreover, a  - 118 good c o n t a c t was  -  d e s i r e d between the r o o t p l u n g e r s  When measuring the q u a l i t y f a c t o r of a one Dee c o n t a c t had  was  the f l u x guide s h o r t s .  r e s o n a t o r a good  the ground arm  observed  t h a t almost  l a t e r present  i t was  always due  no energy was  l e a k i n g out of the Dee when i t  to i n s u f f i c i e n t  to the gaps between the r e s o n a t o r p a n e l s . t u n i n g of the Dee The  caused  virtual  tips.  a c c u r a t e l y a l i g n e d and when good c o n t a c t s were ensured.  frequency.  electrical  to be p r o v i d e d between the grounded p l a t e ( s i m u l a t i n g the  n o d a l plane) and I t was  and  I f any  leakage  c o n t a c t s at the r o o t r a t h e r than  N e i t h e r the coarse nor the  fine  measurable changes i n the Q at the fundamental  c o u p l i n g between the r e s o n a t o r and  the o s c i l l a t o r had  to be  at an a b s o l u t e minimum i n o r d e r to o b t a i n c o r r e c t measurements.  A loose  c a p a c i t i v e c o u p l i n g p l a c e d at the main c o u p l i n g loop opening was  employed.  1.2  Resonator The  found  to be 23.10  f r e q u e n c i e s o f the assembled and MHz  and 69.29 MHz.  d e v i c e s on the frequency, o s c i l l a t o r and  The Due  l e v e l l e d Dee  were  To minimize the e f f e c t of the measuring  a v e r y l o o s e c a p a c i t i v e c o u p l i n g both between the  the r e s o n a t o r was  o v e r a l l frequency  counter  ( e v e n t u a l l y the  used.  s t a b i l i t y of the p r e s e n t r e s o n a t o r was  about  1/10\  to i n h e r e n t q u a l i t i e s of the hot arms, the temperature g r a d i e n t s i n the  transients.  tip.  thoroughly  the r e s o n a t o r and between the frequency  resonator panels  (T  kept  Frequencies  resonant  scope) and  was  caused  A uniform  = 70 °F, T  2  = 150  mechanical  d i s t o r t i o n s d u r i n g the  temperature  change i n temperature o f the hot arm by AT = 80 °F) r e s u l t e d i n almost  1.2  cm d e f l e c t i o n of the hot  The p r e s s u r e of the c i r c u l a t i n g hot water was  d e f l e c t i o n at s e v e r a l p o i n t s i t was  noted  d e f l e c t e d hot arm by a p a r a b o l a was  justified.  d e f l e c t i o n o f the hot arm  35 p s i .  By measuring  t h a t the approximation  t i p of about Ag = 1 cm  °F  of  the  C a l c u l a t i o n shows t h a t a (parabolic  approximation)  arm the  - 119 causes  -  the frequency to change by more than 0.8%.  s h i f t s proved  to be troublesome  d u r i n g the h i g h power t e s t s .  e s p e c i a l l y t r u e d u r i n g the s t a r t - u p procedure. approximation  Ag/'g  <* AT/T  These l a r g e  In the f i r s t  frequency This  order  and so was A f / f .  B e s i d e the temperature  another  f a c t o r was  a l s o i n f l u e n c i n g the  namely the p r e s s u r e i n the c o o l i n g c h a n n e l s . By p r e s s u r a z i n g the system  a r e l a t i v e l y l a r g e frequency change o c c u r r e d .  s m a l l d e f l e c t i o n s of the hot arm pressure  was  T h i s was  frequency,  cooling  a g a i n due  to  caused by changes i n the c o o l i n g water flow  (expansion or c o n t r a c t i o n o f the c o o l i n g c h a n n e l s ) . The measurement  showed t h a t a change i n p r e s s u r e by Ap = +30  p s i r e s u l t e d i n Af = -0.4%.  Random f l u c t u a t i o n s of the water p r e s s u r e and water v e l o c i t y  excited  mechanical v i b r a t i o n s o f the hot arms at t h e i r n a t u r a l mechanical The measured frequency s h i f t s can not be compared w i t h t h e o r e t i c a l because the combined e f f e c t o f the weight  frequency. calculation  of the p a n e l and the p r e s s u r e i n the  channels r e s u l t e d i n e n t i r e l y d i f f e r e n t d i s t o r t i o n s of lower and upper hot arms. The measurement done w i t h one Dee frequency s t a b i l i t y water was  a t the fundamental  at medium power l e v e l showed t h a t frequency was  p r e s e n t i n the c o o l i n g c h a n n e l s . The  magnitude lower when c i r c u l a t i n g the water. due  to temperature  noted t h a t  instabilities achieved  d i f f e r e n c e between the hot arm s k i n and the s u p p o r t i n g  However i n s t a b i l i t i e s  continued.  an o r d e r of  changes d i s a p p e a r e d as soon as e q u i l i b r i u m was  ( i . e . no temperature I-beams).  4 p a r t s i n 10^ when no  s t a b i l i t y was  I t was  the  To minimize  caused by c i r c u l a t i n g the c o o l i n g water  the i n s t a b i l i t i e s ,  spacers were l a t e r i n s e r t e d between  the upper and lower hot arms.  1.3  F i n e Frequency The  Tuning  t u n i n g o f the system by means of t u n i n g b e l l o w s mounted on r o o t  p i e c e s was  t e s t e d both a t low and h i g h power l e v e l s .  T h i s t u n i n g i s y e t to be  - 120  -  t e s t e d at h i g h power l e v e l d u r i n g a c t u a l f i x e d  frequency  l a r g e spread i n the measured v a l u e s was  caused  by an i n a c c u r a t e r e a d i n g of  air  The  p r e s s u r e i n the a c t u a t i n g b e l l o w s .  The  p r e s s u r e d i f f e r e n c e was  bellows Fig.  T a b l e XVIII p r e s e n t s  changes when one Dee was  17).  the  s u b s t i t u t e d f o r changes i n the p o s i t i o n of t u n i n g  A d i s c r e p a n c y i s due s u r f a c e was  the  mounted i n the frame.  i n o r d e r to compare the computed v a l u e s w i t h the measured ones  the b e l l o w s  (see  to the assumption t h a t the c u r r e n t d e n s i t y on  taken to be the same as t h a t a t the r o o t . Measurements  on h a l f - s c a l e r e s o n a t o r s showed t h a t the f i e l d was  Quite a  zero p o s i t i o n r e f e r s to the  s i t u a t i o n when the p r e s s u r e d i f f e r e n c e equals z e r o . measured v a l u e s of frequency  operation.  s m a l l e r than t h a t at the r o o t .  d e n s i t y at the b e l l o w s  A l s o , the f i e l d  surface  d e n s i t y depended on  p o s i t i o n of the b e l l o w s w i t h r e s p e c t to the r o o t p l a n e .  The  quality  the  factors  were u n a f f e c t e d .  1.4  Coarse Frequency A p r o v i s i o n was  probe mechanisms. T a b l e XIX.  The  Tuning made to d e f l e c t the ground arm  The  r e s u l t s of r e s o n a t o r coarse t u n i n g are p r e s e n t e d  l e n g t h of the d e f l e c t e d t i p was  c e n t r e of the a c c e l e r a t i n g gap. Fig. due  20.  The  arm  computed v a l u e s are shown i n  t h i r d harmonic q u a l i t y f a c t o r v a r i e d d u r i n g the t e s t .  d u r i n g the measurement.  in  35 i n . measured from the  The measured and  to a poor c o n t a c t between the ground arm  through  t i p s by means of v o l t a g e  t i p s and  This  was  the grounded p l a t e  In a d d i t i o n some energy c o u l d have been l e a k i n g  the openings which o c u r r e d next  to the f l u x guides d u r i n g the ground  tip deflection. However, coarse frequency  t u n i n g o b t a i n e d by d e f l e c t i n g  t i p s can not be c o n s i d e r e d as the f i n a l problem f o r a number of reasons.  the ground  s o l u t i o n of the frequency  Besides  the mechanical  problems w i t h r e g a r d to RF o p e r a t i o n concern  arm  tuning  problems, the main  the t h i r d - t o - f i r s t  harmonic  - 121 -  TABLE XVIII Measured frequency s h i f t s P.  p  Aft  f  (mm)  ol (MHz)  (%)  1  O  2  (lb/in.)  2  (lb/inl)  caused by t u n i n g b e l l o w s A  f  o l,  f  Af  o3  f o3  (MHz)  03  It  01  (%)  10  0  - 6  23.0994  +0.0147  69.2291  + 0.0136  2.997  10  2.5  - 5  23.0987  + 0.0117  69.2276  + 0.0114  2.997  10  5  - 4  23.0980  + 0.0087  69.2258  + 0.0088  2.997  10  7.5  - 3  23.0972  + 0.0052  69.2223  + 0.0038  2.997  10  10  - 2  23.0963  + 0.0013  69.2206  + 0.0013  2.997  10  12.5  - 1  23.0953  -0.0030  69.2172  -0.0036  2.997  10  15  0  23.0947  -0.0056  69.2160  -0.0053  2.997  10  0  23.0960  0  69.2197  0  2.997  TABLE XIX Measured frequency s h i f t s half  Ag  ol (MHz) f  caused by ground Af  arm t i p d e f l e c t i o n  f oi  Af 03  f 03  It 03  01  turns  (mm)  - 3  + 3.81  23.1670  + 0.290  69.2778  + 0.068  2.9903  - 2  + 2.54  23.1454  + 0.196  69.2626  + 0.046  2.9925  - 1  + 1.27  23.1236  + 0.102  69.2476  + 0.024  2.9947  0  23.1000  + 1  - 1.27  23.0771  - 0.099  69.2075  - 0.033  2.9989  + 2  - 2.54  23.0540  - 0.199  69.1917  - 0.056  3.0013  + 3  - 3.81  23.0327  - 0.292  69.1726  - 0.084  3.0032  + 4  -  5.08  23.0159  - 0.364  69.1530  - 0.112  3.0046  + 5  - 6.35  22.9941  - 0.458  69.1284  - 0.148  3.0063  + 6  - 7.62  22.9686.  - 0.569  69.1100  - 0.174  3.0089  0  (%>  0  (MHz)  69.2306  (%)  0  2.9969  - 122 frequency  r a t i o and the f i x e d probe's c a l i b r a t i o n  IV.1.5).  I t i s , however, s t r o n g l y recommended  be r e t a i n e d f o r a d j u s t i n g the frequency  ratio.  (see S e c t i o n s II.4.5 and  t h a t the d e f l e c t i n g mechanism T h i s would be done o n l y once  p r i o r t o h i g h power o p e r a t i o n .  1.5  RF Probes In o r d e r to measure the a c t u a l r e s o n a t o r v o l t a g e d u r i n g c y c l o t r o n  o p e r a t i o n two methods w i l l be used.  C a p a c i t i v e pick-up probes w i l l i n f o r m us  on the l e v e l of the r e s o n a t o r v o l t a g e , an a c c u r a t e v a l u e o f the r e s o n a t o r peak v o l t a g e w i l l be o b t a i n e d by measuring the energy g a i n of p a r t i c l e s u s i n g beam probes. The f i x e d RF probes i n s t a l l e d on e i t h e r Dee were c a l i b r a t e d a g a i n s t a t h e r m i o n i c diode probe d i r e c t l y by t o u c h i n g the t i p of the p a r t i c u l a r hot arm. The c a l i b r a t i o n was done w i t h about 500 V on r e s o n a t o r . l o a d i n g e f f e c t r e s u l t e d i n a frequency frequency  Although  s h i f t o f Af = 80 kHz a t the nominal  of 23.1 MHz, no change i n the v o l t a g e amplitude  Changing the l o a d i n g impedance connected  c o u l d be d e t e c t e d .  to the probe output  d i d n o t have  measurable e f f e c t on the r e s o n a t o r e l e c t r i c a l c h a r a c t e r i s t i c s . r a t i o s were e q u a l t o 1500 a t the middle  the t i p  The  range o f the coarse frequency  adjustment and v a r i e d as the ground arm t i p s were d e f l e c t e d . measurement of the r e s o n a t o r v o l t a g e amplitude  step-up tuning  No p r e c i s e  can be c a r r i e d out u n l e s s the  ground arm t i p s are a t t h e i r o r i g i n a l p o s i t i o n where a c a l i b r a t i o n was done. The c a l i b r a t i o n of the v o l t a g e probes proved two Dees were assembled i n the vacuum tank. presence  to be q u i t e d i f f i c u l t when  The l o a d i n g e f f e c t  and the  of poor RF c o n t a c t s seemed to be the main f a c t o r s which i n f l u e n c e d  the c a l i b r a t i o n .  I t was suspected  t h a t the v i r t u a l n o d a l p l a n e i n the middle  of the a c c e l e r a t i n g gap was b e i n g d i s t o r t e d .  To make the v o l t a g e probes read  the same v o l t a g e , the c a p a c i t i v e p l a t e s had to be r e p o s i t i o n e d so t h a t a l l the  - 123  -  probes were at the same p o s i t i o n w i t h r e s p e c t to the ground arms. c o n n e c t i o n had  to be secured f o r each probe.  Due  to mechanical  A good  v i b r a t i o n s of  the hot arms the v o l t a g e measured d u r i n g h i g h power o p e r a t i o n c o u l d determined The  w i t h an accuracy  of o n l y about  1%.  o t h e r method i s based on a c c u r a t e l y measuring the energy o f  p a r t i c l e s by means of movable scanning probe. of the r e s o n a t o r v o l t a g e amplitude  be  A very accurate  can then be made.  the  determination  During the RF t e s t s  probe c o u l d a l s o be used to measure the v o l t a g e u n i f o r m i t y along the hot tips.  1.6  this arm  T h i s method has y e t to be t e s t e d .  C o u p l i n g Loop Several tests related  to the c o u p l i n g loop s e l f - i n d u c t a n c e , the  t i p to loop v o l t a g e r a t i o and the vacuum s e a l were c a r r i e d The  loop s e l f - i n d u c t a n c e was  out.  measured u s i n g Hewlett-Packard  impedance meter w i t h o n l y the ground arms i n s t a l l e d i n the vacuum Measuring the impedance o f the loop a t two v a l u e s L^ = 0.221  uH and L^ = 0.212  uH.  different  resonator  vector tank.  f r e q u e n c i e s gave the  These were i n v e r y good agreement  w i t h the computed v a l u e L = 0.224 uH. The  r e s o n a t o r t i p to loop v o l t a g e r a t i o was  measured f o r the nominal  p o s i t i o n of the loop w i t h the v e r t i c a l p a r t of the t r a n s m i s s i o n l i n e i n p l a c e . S e v e r a l ways of e x c i t i n g for different  the system have been t r i e d  c o u p l i n g s between the Dee  and  the RF s o u r c e .  p a r t of the t r a n s m i s s i o n l i n e i n p l a c e i t was system at the r e s o n a t o r resonant  frequency.  difficult I t was  t h a t t h i s r a t i o v a r i e d w i t h the d r i v i n g frequency. we  giving different  were q u i t e c e r t a i n t h a t the e x i s t i n g r a t i o was  to  With the  results vertical  e x c i t e the  shown i n S e c t i o n I I . 10.3 Although  this test  failed  v e r y c l o s e to the  computed v a l u e because the p a r a l l e l impedance of a one Dee  r e s o n a t o r measured  w i t h the v e c t o r impedance meter at the c o u p l i n g p o i n t turned out to be  very  - 124  c l o s e to the computed v a l u e . S i n c e the f i r s t  -  T h i s t e s t was  done w i t h one  Dee  only.  attempts to make a good vacuum-tight t e f l o n s e a l d i d not  succeed, the ceramic d i s c was  temporarily replaced with a t e f l o n s e a l .  Most  RF h i g h power t e s t s were done w i t h the t e f l o n s e a l which a t some times presented  many d i f f i c u l t i e s .  The  RF l o s s e s and  s t r e s s e s o r i g i n a t i n g from them  seemed to p r e s e n t  no d i f f i c u l t y .  were p r o b a b l y  to m e c h a n i c a l f o r c e s at the c o u p l i n g loop  1.7  due  Transmission  L i n e and  However, l e a k s developed s e v e r a l times  The  initial  d e s i r e d i n p u t impedance was oscillator.  The  i n the b e g i n n i n g  due  to the l a c k of good  t u n i n g of the l i n e i n o r d e r to a t t a i n a  done u s i n g an o s c i l l o s c o p e and  could e a s i l y  i n p u t impedance, because i n d i v i d u a l resonant  of the system were s e p a r a t e d  by more than a few  frequencies  Jennings c a p a c i t o r s were i n s e r t e d i n the l i n e as i n d i c a t e d by The  but l a t e r removed.  found t h a t the p l a t e - t o - g r o u n d  I t was  t h i r d c a p a c i t o r , CP(6), was  the s t r a y c a p a c i t y i n the a m p l i f i e r c a b i n e t was  t h i r d c a p a c i t o r had  almost no e f f e c t on t u n i n g  initially capacity  particularly  The  the Jennings c a p a c i t o r s were o b t a i n e d  the  input This  The  from t e s t s at a  was  charts low  The measurement done at TRIUMF showed that at h i g h f r e q u e n c i e s  capacitors represented  a capacitance  i n s e r i e s with  magnitude depended on the p a r t i c u l a r arrangement and measurement.  that  capacitor  the computed v a l u e s .  t r u e f o r the c a p a c i t o r i n the middle of the l i n e .  supplied with frequency.  i n disagreement w i t h  installed  of such a v a l u e  the l i n e .  the  together  s e t t i n g s of the Jennings c a p a c i t o r s which r e s u l t e d i n the r e q u i r e d impedance seemed to be  Our  c o n c l u s i o n was  be  MHz.  computer program RESLINE.  with  the Hewlett-Packard  t r a n s m i s s i o n l i n e w i t h a dummy l o a d at the end  tuned f o r the proper  Two  assembly.  Resonances of the System  Some problems were e x p e r i e n c e d measuring equipment.  and  an inductance RF  whose  contacts during  t h a t at h i g h f r e q u e n c i e s  the  the c a p a c i t o r  the values  -  125  -  TABLE XX Measured and computed resonances f o r a one Dee-resonant l i n e system f computed (MHz)  Type  f measured (MHz)  Type s e r i e s or parallel  5.879  P  6.705  P  11.285  S  13.020  S  18.818  P  19.406  P  22.585  S  22.556  S  22.660  P  22.760  P  23.826  s  23.666  S  24.983  P  24.327  P  48.380  s  24.390  S  54.870  p  25.290  P  58.450  s  35.820  S  59.36  p  75.616  P  S-P  75.739  S  88.340  P  89.884  S  68.000  were a l i t t l e h i g h e r  than those  i n d i c a t e d on the c h a r t s .  The t r a n s m i s s i o n l i n e t u n i n g became more d i f f i c u l t when a one Dee resonator  r e p l a c e d the dummy l o a d .  However, most t r o u b l e s d i s a p p e a r e d  when the  v e c t o r impedance meter was employed to measure the i n p u t impedance o f the l i n e , and  the accuracy  particular, T a b l e XX.  of the measured q u a n t i t i e s s i g n i f i c a n t l y  the resonant  increased.  In  f r e q u e n c i e s o f the system c o u l d be determined, see  I t was a l s o proved t h a t any d e s i r e d v a l u e o f i n p u t impedance ( a t the  fundamental frequency) c o u l d be a t t a i n e d by a s u i t a b l e combination o f the two  -  126 -  TABLE XXI Measured and computed resonances f o r a two Dee-resonant l i n e system f computed (MHz)  capacitors. resonances  Type  22.622  S  22.699  S  22.660  P  22.937  P  23.570  S  23.212  S  23.820  P  23.489  P  24.070  s  23.619  S  24.920  p  24.532  I t was  noted t h a t some  resonances  •  •  -P  o f the system  coincided with  o f the power a m p l i f i e r elements, which o f c o u r s e , caused  o s c i l l a t i o n s a t those f r e q u e n c i e s . could  f measured (MHz)  Type  not  The o r i g i n a l l y used narrow band  parasitic amplifier  cope w i t h problems a r i s i n g from an u n s t a b l e resonant l o a d a t the  end o f l i n e . The  above mentioned t e s t s were l a t e r r e p e a t e d when a new t e t r o d e a m p l i f i e r  r e p l a c e d the t r i o d e narrow band a m p l i f i e r . onances  S i m i l a r r e s u l t s were found. The r e s -  o f the two Dee r e s o n a t o r - t r a n s m i s s i o n l i n e system were measured i n  the range between 20 MHz and 25 MHz. p r e s e n t e d i n T a b l e XXI.  Both the computed and measured v a l u e s a r e  The c a l c u l a t i o n o f the resonances was done w i t h the  r e s o n a t o r frequency e q u a l t o 2 2 . 6 6 MHz. The  c u r r e n t and v o l t a g e d i s t r i b u t i o n s i n the s e c t i o n o f the l i n e between  the r e s o n a t o r and the middle computed v a l u e s . computed v a l u e . middle  c a p a c i t o r CP(3)  turned out t o be v e r y c l o s e to  The v o l t a g e node p o s i t i o n was found to be w i t h i n 3 cm o f the However, the c o n d i t i o n s i n the s e c t i o n o f the l i n e between the  c a p a c i t o r and the t e t r o d e a m p l i f i e r were e n t i r e l y d i f f e r e n t .  T h i s was  - 127 accounted f o r by a v e r y h i g h c a p a c i t y ( p l a t e - t o - g r o u n d  i n p a r a l l e l with a stray  c a p a c i t y ) which was connected a t t h e i n p u t o f t h e l i n e .  Due t o t h e e x c e s s i v e  amount o f s t r a y c a p a c i t y c a p a c i t o r s CP(2)  and, i n p a r t i c u l a r , CP(3), had t o be  r e a d j u s t e d t o g i v e a c o r r e c t i n p u t impedance.  1.8  Comparison o f R e s u l t s from Lumped and D i s t r i b u t e d Parameter Representations Many c a l c u l a t i o n s such as t h e i n p u t impedance o f t h e system v s frequency,  r e s o n a t o r v o l t a g e response v s frequency,  beam l o a d i n g e t c . , c o u l d only be  c a r r i e d out i f our r e a l system w i t h d i s t r i b u t e d parameters was r e p l a c e d w i t h lumped c o n s t a n t s .  The lumped parameters were computed a t t h e resonant  frequency  of t h e r e s o n a t o r and i t was, t h e r e f o r e , n e c e s s a r y  frequency  range where they s t i l l  represented  t o determine the  our d i s t r i b u t e d system.  The  e a s i e s t check o f v a l i d i t y o f lumped c o n s t a n t s was t o d i r e c t l y measure the r e s o n a t o r v o l t a g e and i n p u t impedance as f u n c t i o n s o f frequency them w i t h  and compare  r e s u l t s o f computer code MATCH and w i t h r e s u l t s o b t a i n e d  from a  lumped parameter r e p r e s e n t a t i o n o f our r e s o n a t o r - t r a n s m i s s i o n l i n e  system  (computer program LUMP). An i n v e s t i g a t i o n was c a r r i e d out both w i t h one Dee r e s o n a t o r and two Dee r e s o n a t o r connected a t the end o f resonant  line.  The i n p u t impedance was  measured u s i n g the Hewlett-Packard v e c t o r impedance meter.  The r e s o n a t o r  v o l t a g e was d i r e c t l y measured on the T e c t r o n i x scope w i t h a s i g n a l taken v o l t a g e probe a t t h e r e s o n a t o r  tip.  from the  When measuring t h e v o l t a g e dependence a  r e s i s t o r s i m u l a t i n g t h e o p e r a t i n g r e s i s t a n c e of the a m p l i f i e r was connected i n s e r i e s with the o s c i l l a t o r . r e s i s t a n c e a t t h e resonant  The i n p u t impedance was matched t o t h e o p e r a t i n g frequency  of t h e r e s o n a t o r .  F i g s . 39, 40 show the  r e s u l t s from t h e t e s t s when two Dees were l o a d i n g the t r a n s m i s s i o n Comparison w i t h  line.  the computed v a l u e s p l o t t e d i n F i g s . 39, 40 a l s o r e v e a l e d t h a t  - 128 -  the r e p r e s e n t a t i o n o f the system by t h r e e lumped parameter resonant c i r c u i t s was j u s t i f i e d  i n the range  <f  Q  - Af, f  - Af>, Af b e i n g e q u a l t o 5 kHz.  was mainly due to a r e p r e s e n t a t i o n o f a resonant l i n e by a s i n g l e parameter resonant c i r c u i t .  In t h i s range  the measured q u a n t i t i e s  This  lumped roughly  f o l l o w e d the computed ones. From the curves i n F i g s . of the whole system  39, 40 one can a l s o e s t i m a t e the q u a l i t y  c o n s i d e r e d as a s i n g l e p a r a l l e l resonant c i r c u i t .  q u a l i t y f a c t o r o f the r e s o n a t o r i s a p p r o x i m a t e l y e q u a l to the q u a l i t y of  the whole system  factor  (no impedance connected  The factor  at the l i n e i n p u t ) o r t w i c e as  h i g h as the loaded q u a l i t y f a c t o r when the r e s i s t o r s i m u l a t i n g the tube resistance  i s connected  at the i n p u t o f l i n e .  v o l t a g e c u r v e , F i g . 40, y i e l d s Q  = 3250, i . e . Q = 6500. Li  F i g . 39, g i v e s Q = 6050.  F o r a two Dee r e s o n a t o r the The impedance c u r v e ,  - 129 2.  HIGH POWER LEVEL 2.1  Sparking  and M u l t i p a c t o r i n g  High power t e s t s s t a r t e d w i t h a Dee mounted i n an a u x i l i a r y frame. Dee was e n e r g i z e d t o a l e v e l of 40 t o 50 kV i n a i r , where sparkovers r e s o n a t o r t i p s prevented sparks  the attainment  of higher voltages.  The  at the  A t the 50 kV l e v e l  o c c u r r e d at the r e s o n a t o r t i p s i n the upper segments of the r e s o n a t o r ,  mainly between the h o t arm-ground arm t i p s and a c r o s s the ground arm-ground arm f l u x guide gap. the v o l t a g e  No sparks were observed  i n the lower  r e s o n a t o r segments o r at  probes.  An improvement was achieved when t h e r e s o n a t o r was e n c l o s e d i n p l a s t i c sheets  and f i l l e d up w i t h an a i r - f r e o n m i x t u r e .  i n c r e a s e d to 72 kV when d i s c h a r g e s o c c u r r e d  The r e s o n a t o r v o l t a g e c o u l d be  again.  Power l o s s i n the r e s o n a t o r was measured c a l o r i m e t r i c a l l y u s i n g two t h e r m i s t o r s a t t a c h e d to the i n l e t and o u t l e t c o o l i n g channels.  The r e s u l t was  w i t h i n 5% o f the computed v a l u e f o r 72 kV v o l t a g e l e v e l i n r e s o n a t o r . Resonator o p e r a t i o n under h i g h vacuum brought along the problems associated with multipactoring.  With the p r e s s u r e o f the o r d e r o f magnitude  -5 10  T o r r a t the s t a r t of the t e s t s and r e s o n a t o r panels n o t g i v e n any s p e c i a l  c l e a n i n g , the r e s o n a t o r v o l t a g e c o u l d n o t be made to exceed a l e v e l o f 170 V f o r a long p e r i o d .  T h i s lower  threshold multipactoring voltage  corresponded  v e r y c l o s e l y t o the c a l c u l a t e d v a l u e o f 180 V (see S e c t i o n I I . 3 . 3 ) . time a b l u e glow d i s c h a r g e was observed  a t the r e s o n a t o r t i p s .  No d e t a i l e d  a n a l y s i s o f the v o l t a g e wave form d u r i n g m u l t i p a c t o r i n g was done. a m p l i f i e r was then p u l s e d b u t i t was not u n t i l  At t h i s  The  the p r e s s u r e was improved to  -6 10  T o r r t h a t we c o u l d break through m u l t i p a c t o r i n g and the r e s o n a t o r was  operated  at high voltage  level.  For the next s e t o f measurement the r e s o n a t o r p a n e l s were c l e a n e d u l t r a s o n i c a l l y , i n s t a l l e d i n t o the vacuum tank and baked a t about 200 °F.  - 130 The  p r e s s u r e of the o r d e r o f magnitude 10  6  was then e a s i l y o b t a i n e d .  problems due t o m u l t i p a c t o r i n g were expected  from t h e t h e o r e t i c a l c a l c u l a t i o n s  of t h e v o l t a g e r i s e time o f a two Dee r e s o n a t o r w i t h r e g a r d t o the c l e a n i n g o f the p a n e l s .  Fewer  and from the p r e c a u t i o n s  Although  taken  i t seemed t h a t m u l t i -  p a c t o r i n g o c c u r r e d at low v o l t a g e s i t c o u l d not be c l e a r l y i d e n t i f i e d .  With a  good broadband t e t r o d e a m p l i f i e r on hand and the a b i l i t y  into  resonator few  t o put 140 kW  ( i n p u l s e ) i t was always p o s s i b l e t o get to h i g h e r v o l t a g e s w i t h i n a  seconds. To  check the l e v e l o f the breakdown v o l t a g e i n the CRM,  gap was reduced  t o 2 inches from the nominal 6 i n c h e s .  (240 kV a c c e l e r a t i n g v o l t a g e ) , no sparks  the a c c e l e r a t i n g  At t h e 120 kV l e v e l ,  occurred, provided  t h a t the vacuum  -6 was b e t t e r than 5 x 10  Torr.  Sparks were u s u a l l y e x p e r i e n c e d  i n the b e g i n n i n g  o f h i g h power t e s t s .  A f t e r a p e r i o d o f c o n d i t i o n i n g i t was no problem t o h o l d 100 kV on r e s o n a t o r f o r s e v e r a l hours.  However, a t times  c o n t a c t s a t the r o o t .  sparks  o c c u r r e d owing t o f a i l u r e s i n  E v e n t u a l l y , the poor c o n t a c t s at t h e r o o t were  r e s p o n s i b l e f o r RF leakage which gave r i s e t o RF d i s c h a r g e s r e s u l t i n g i n a b l u e glow everywhere i n s i d e the vacuum tank. With a good vacuum s e a l n e i t h e r s p a r k i n g n o r m u l t i p a c t o r i n g o c c u r r e d around the c o u p l i n g l o o p .  In one i n s t a n c e i o n s p u t t e r i n g was i n i t i a t e d by a  s m a l l l e a k i n the vacuum s e a l r e s u l t i n g i n l a r g e m e t a l d e p o s i t s on t h e t e f l o n d i s c and on the i n n e r conductor  j u s t above the s e a l .  Heat d i s s i p a t i o n  caused  a temperature r i s e under which the t e f l o n s e a l s o f t e n e d and was bent inwards. The  c u r r e n t c a r r y i n g f i n g e r s underneath t h e i n s u l a t o r l o s t o f f i n g e r s f o l l o w e d and t h i s l e d  contacts,  small  sparks  and b u r n i n g  seal.  The e f f e c t of t h e magnetic f i e l d o f the magnet on the RF o p e r a t i o n was  a l s o i n v e s t i g a t e d and found  undetectable.  to the d e s t r u c t i o n of the  - 131  2.2  RF  -  Contacts  To o b t a i n good RF c o n t a c t s r e c t a n g u l a r s p r i n g s (Melrose, fingerstocks  (Eimac, #CF-800 and  #5024/0431/750) were employed withstand  (see F i g . 71).  severe m e c h a n i c a l and  s p r i n g s and  f i n g e r s t o c k s was  i m p e r f e c t RF  CF-900) and  fuzz-buttons  (Tecknit,  At h i g h power l e v e l they had  e l e c t r i c a l stresses.  u s u a l l y due  #60189),  The  damage to both  to s p a r k i n g which was  to  the  caused by  contact.  A v e r y poor RF c o n t a c t e x i s t e d o r i g i n a l l y between the f l u x guide s e c t i o n s and  the r o o t p i e c e s .  between the f l u x  guides  and  the m e c h a n i c a l misalignments  the r e s o n a t o r p a n e l s  p r o p e r t i e s of the r e s o n a t o r . out of the r e s o n a t o r due  Both t h i s and  end  i n f l u e n c e d the e l e c t r i c a l  At h i g h power l e v e l the amount of energy l e a k i n g  to poor c o n t a c t s at r o o t was  a result  d i s c h a r g e s o c c u r r e d behind  when new  flux  the r e s o n a t o r .  guides were i n s t a l l e d .  the good e l e c t r i c a l c o n t a c t was  quite s i g n i f i c a n t  The  both  as  c o n t a c t s were improved  I t s h o u l d be p o i n t e d out  maintained  and  t h a t as l o n g  as  s p r i n g s and f i n g e r s t o c k s  performed s a t i s f a c t o r i l y . Fuzz-buttons the hot arm sparks  and  l a t e r s l i d i n g s p r i n g s p r o v i d e d a good RF c o n t a c t between  t i p s and between the hot arm  at the t i p s was  observed.  t i p s d u r i n g the coarse frequency  t i p and  f l u x guide.  No  damage due  To enable v e r t i c a l motion of the ground  2.3  and had  Voltage  and  Owing to m e c h a n i c a l s t r e s s e s e x e r t e d upon the c o u p l i n g loop  the f i n g e r c o n t a c t s between the loop and times  arm  t u n i n g , S-shaped p i e c e s of 0.005 i n . t h i c k  b r a s s sheet were used to p r o v i d e a c o n t a c t between the beam probe housing the ground arms.  to  to be  the ground arm were damaged s e v e r a l  replaced.  and Frequency  In the b e g i n n i n g  i t was  Stability found  difficult  to f e e d RF power i n t o  the  - 132 r e s o n a t o r f o r two  -  reasons: m u l t i p a c t o r i n g and  spacers between the hot arms the frequency than 1% when the m u l t i p a c t o r i n g range was resonator. necessary  temperature t r a n s i e n t s .  of the r e s o n a t o r v a r i e d by f a r more passed  and power was  S i n c e the o r i g i n a l t r i o d e a m p l i f i e r was to tune and match the  amplifier  o r d e r to punch through m u l t i p a c t o r i n g . disappeared  a d i f f e r e n t impedance was  Consequently,  the a m p l i f i e r had  to be  presented retuned  s h i f t s due  panels.  r e q u i r e d and  to the  and  the c y c l e was  position.  T h i s improved the s t a r t up and enabled  v o l t a g e l e v e l , however, the i n s t a b i l i t i e s  No  detailed  frequency  and  repeated.  was  This resonator  The  only  thus l o c k the hot arms i n t h e i r the attainment  ( A f / f = ±0.01%) due  and v o l t a g e s t a b i l i t y  of a h i g h e r  to water p r e s s u r e  a n a l y s i s was  made at  never p o s s i b l e to run the r e s o n a t o r under  c o n d i t i o n s f o r a long  New  More power  p s i ) remained.  stage because i t was steady  rematched.  to thermal d e f l e c t i o n s of the  to i n s e r t spacers i n the beam gap  (±1/2  always  amplifier.  t h e r e f o r e , developed.  remedy was  fluctuations  narrowband, i t was  Once the m u l t i p a c t o r i n g l o a d  r e s u l t e d i n l a r g e frequency r e t u n i n g was  f e d to the  to the i o n l o a d i n g c o n d i t i o n s i n  then f e d to the r e s o n a t o r and more heat was,  A new  With no  this  relatively  time.  t e s t s w i t h an improved c o o l i n g system and w i t h the t e t r o d e a m p l i f i e r  broadband n e u t r a l i z e d proved  to be more s u c c e s s f u l l .  p a n e l s were c l e a n e d i n a 10% water s o l u t i o n of #33 u l t r a s o n i c c l e a n i n g tank b e f o r e i n s t a l l i n g  S i n c e the  resonator  O a k i t e i n a wooden  them i n t o the vacuum tank t h e r e —6  were no problems i n a c h i e v i n g good vacuum of the o r d e r o f magnitude 10 T h i s f a c t o r and  the f a c t  t h a t a 3 i n c h gap was  r e s o n a t o r ) i n f l u e n c e d the s t a r t up procedure. amplitude  and width  of a p u l s e was  no  longer present  (a two  Torr. Dee  A pulsing device with a v a r i a b l e  used to break through m u l t i p a c t o r i n g . Once m u l t i p a c t o r i n g was  passed  No  d i f f i c u l t i e s were e x p e r i e n c e d  t h i s time.  and  the r e s o n a t o r v o l t a g e reached  about 20 kV a s w i t c h to a s e l f - o s c i l l a t o r y mode  - 133 -  occurred.  Now the r e s o n a t o r v o l t a g e d e t e c t o r output  reference voltage  and  the  e r r o r s i g n a l was a m p l i f i e d t o d r i v e e i t h e r the  i n p u t modulator or the s c r e e n modulator. the v o l t a g e l e v e l f i n a l value.  The i n p u t modulator was employed when  i n the r e s o n a t o r was t o be i n c r e a s e d from about 20 kV t o i t s  The s c r e e n modulator, however, was used to m a i n t a i n  v o l t a g e amplitude a t a g i v e n l e v e l . frequency  was compared w i t h a  At approximately  a steady RF  100 kV l e v e l the  s t a b i l i t y was ±0.01% and the v o l t a g e s t a b i l i t y was ±0.01%.  I t s h o u l d be noted t h a t s i n c e the automatic frequency  c o n t r o l was not y e t  i n o p e r a t i o n the power a m p l i f i e r had t o be d r i v e n by a s i g n a l d e r i v e d from the resonator  a l l the time.  resonator  frequency.  The RF system was thus s e l f - o s c i l l a t o r y  T h i s allowed  considerable resonator  At h i g h v o l t a g e l e v e l when the r e s o n a t o r  frequency  drift  a t the d u r i n g warm-up.  has s t a b i l i z e d  the a m p l i f i e r  w i l l be d r i v e n by a s y n t h e s i z e r .  2.4  C o u p l i n g Loop and T r a n s m i s s i o n  Line  I t was found t h a t the c o u p l i n g loop c o u l d be adequately  c o o l e d by t h e  a d d i t i o n of a c o o l i n g channel around the h o r i z o n t a l edge o f the l o o p .  However,  the v e r t i c a l p a r t o f the t r a n s m i s s i o n l i n e next t o the c o u p l i n g loop was warm due  to the c o n d u c t i o n  coupling loop. to  of t h e heat developed i n the v e r t i c a l  s e c t i o n of the  The l o s s e s i n the t r a n s m i s s i o n l i n e s e c t i o n near the loop due  the i n p u t c u r r e n t were s m a l l enough i n t h e CR c y c l o t r o n b u t they  will  i n c r e a s e by a f a c t o r o f 100 i n the main c y c l o t r o n i f the same dimensions a r e used  (see S e c t i o n Due  II.7.2).  to a lower q u a l i t y f a c t o r o f the r e s o n a t o r  and because the o u t e r  conductor o f the l i n e was made o f aluminum, the power l o s s i n t h e p a r t of the l i n e between t h e loop and the middle c a p a c i t o r might have been h i g h e r by a f a c t o r o f 1.5 than the computed v a l u e o f 500 W.  A l s o because the c a p a c i t a n c e  - 134 -  around the tube was v e r y h i g h , the c a p a c i t o r s , e s p e c i a l l y the m i d d l e one, were set  to s l i g h t l y d i f f e r e n t values  thereby  lowering  the s t a n d i n g wave r a t i o i n  the l i n e near the tube but i n c r e a s i n g the s t a n d i n g wave r a t i o i n the f i r s t and  second s e c t i o n s o f the l i n e .  F r e s h a i r was l a t e r blown i n the t r a n s m i s s i o n  l i n e i n o r d e r t o reduce h e a t i n g of the l i n e i n t h e f i r s t  and second s e c t i o n s .  L a t e r when the l i n e was taken a p a r t , l a r g e d e p o s i t s o f oxide were found the v o l t a g e node.  near  Depending on the d e n s i t y of the c o l o r i n g , the node was  determined t o be w i t h i n 3 cm o f t h e computed v a l u e . S e v e r a l sparks o c c u r r e d i n the t r a n s m i s s i o n l i n e near the middle c a p a c i t o r . These sparks were always due to m o i s t u r e  and water drops o r i g i n a t i n g from the  c o o l i n g l i n e s f o r the m i d d l e c a p a c i t o r .  Otherwise no problems were  experienced  w i t h t h e t r a n s m i s s i o n l i n e d u r i n g h i g h power t e s t s . 2.5  RF Power A m p l i f i e r The  o r i g i n a l d e s i g n proposed the use o f a ML-7560 t r i o d e o p e r a t i n g i n s e l f -  b i a s e d , grounded cathode c o n f i g u r a t i o n . n e u t r a l i z a t i o n d i d n o t perform onant l o a d . frequency,  coil  s a t i s f a c t o r i l y when loaded by an u n s t a b l e  res-  Since t h i s form o f n e u t r a l i z a t i o n was e f f e c t i v e a t o n l y one the resonant  r e s o n a t o r frequency oscillations. The  This a m p l i f i e r u t i l i z i n g  n e u t r a l i z i n g c i r c u i t had t o be r e a d j u s t e d each time t h e  changed.  I n a d d i t i o n , t h e r e were problems w i t h  parasitic  Consequently, a new d e s i g n was r e q u i r e d .  present  CRM RF a m p l i f i e r , designed  and b u i l t by C o n t i n e n t a l E l e c t r o n i c s , 18  employes the Eimac 4CW250,000 t e t r o d e i n a grounded cathode c o n f i g u r a t i o n . The  single-ended  oscillations  g r i d n e u t r a l i z a t i o n used reduces the p o s s i b i l i t y o f p a r a s i t i c  f o r a wide band o f f r e q u e n c i e s . 18  The  RF a m p l i f i e r  amplifiers, grid  f o r the main c y c l o t r o n RF system c o n s i s t s o f f o u r power  each u s i n g a p a i r o f 4CW250,000 t e t r o d e s i n a p u s h - p u l l grounded  circuit.  - 135 CHAPTER V. The  experimental  -  SUMMARY AND  CONCLUSIONS  t e s t s c a r r i e d out to date have shown t h a t the  c h a r a c t e r i s t i c s are v e r y c l o s e to the p r e d i c t e d ones.  resonator  The measured  resonator  parameters such as the power l o s s , the r e s o n a t o r lumped c a p a c i t y and s e l f - i n d u c t a n c e were w i t h i n 5% of the computed v a l u e s . measured on a one Dee two  resonator  Good q u a l i t y f a c t o r s were  (CRM)rQ^ = 6300 (7100), Q  3  = 9000 (12400).  Dees were assembled, the fundamental Q remained the same, but  harmonic Q dropped to about 6200. the power l o s s e s were found  The  the loop  q u a l i t y f a c t o r s , resonant  the  When  third  frequencies  to be s i g n i f i c a n t l y i n f l u e n c e d by alignment  and  e r r o r s and  by poor c o n t a c t s between the r e s o n a t o r segments. S e v e r a l methods of r e s o n a t o r frequency at h a l f - s c a l e .  bellows  a t the r o o t , s a t i s f i e d the c r i t e r i a p l a c e d on the v o l t a g e v a r i a t i o n  (<3%).  Although  (<.5%  two  of them, ground arm  on modelled  onators  the a c c e l e r a t i n g gap  Only  t u n i n g were v e r i f i e d  t i p d e f l e c t i o n and  i n the c e n t r a l r e g i o n ) and q u a l i t y f a c t o r  a t u n i n g method by means of a ground arm  n o r m a l l y be p r e f e r r e d because o f l a r g e r frequency  res-  tuning along  variation  t i p d e f l e c t i o n would  range, f o r e n g i n e e r i n g  reasons  and because of p o s s i b l e problems w i t h the RF f l a t - t o p o p e r a t i o n , t u n i n g u s i n g bellows  a t the r o o t was  motion of Aft - -.25 t e s t s have proved  adopted.  i n . was  ( A f ^ ±.02%) w i l l  achieved when running  summarized i n T a b l e I .  the r e s u l t i n g h e a t i n g .  then be accomplished T h i s i s y e t to be  s t a b i l i t y was The  r e s o n a t o r hot arms caused by A new  f o r a bellows  (CRM).  High power  by  Automatic  i n c l u d i n g the b e l l o w s  tested.  The v o l t a g e  a l s o ±.01%.  The  i n s t a b i l i t i e s were m a i n l y  as  frequency system  stability  the RF system i n s e l f - o s c i l l a t o r y mode at the 100  ±.01%, the frequency  variations.  +.02%  t h a t the t u n i n g b e l l o w s system i s designed m e c h a n i c a l l y so  i n the RF feedback path.  was  change of A f ^  measured on p r o t o t y p e r e s o n a t o r s  to w i t h s t a n d h i g h c u r r e n t s and control  A frequency  the  kV  level  d e s i r e d t o l e r a n c e s are  due  to d e f l e c t i o n s of  the  temperature changes and by c o o l i n g water p r e s s u r e  d e s i g n of the r e s o n a t o r p a n e l s - f l o a t i n g s k i n d e s i g n - w i l l  be used i n the main c y c l o t r o n (the d e f l e c t i o n due  to temperature changes  reduced  - 136 by few  a f a c t o r of 10).  Although  the c o o l i n g system has a l r e a d y been improved, a  a d d i t i o n a l improvements may be r e q u i r e d i n o r d e r to reduce the p r e s e n t water  pressure fluctuations  (±.5 p s i ) .  The d e s i g n o f a coarse frequency  change o f 3%  ( s h i f t o f a r o o t plane) was abandoned due to e n g i n e e r i n g problems and a v e r y tight  time s c h e d u l e .  Tuning  diaphragms, d e s i r e d d u r i n g RF f l a t - t o p o p e r a t i o n i n  fo3 o r d e r t o a t t a i n an exact frequency  ratio  = 3 , were s u c c e s s f u l l y  t e s t e d a t low  oi power l e v e l on a r e s o n a t o r modelled  at f u l l - s c a l e .  No f i n a l d e c i s i o n r e g a r d i n g  the use of the diaphragms i n e i t h e r the CR c y c l o t r o n o r main c y c l o t r o n has y e t been made. M u l t i p a c t o r i n g p r e s e n t e d many problems d u r i n g h i g h power o p e r a t i o n w i t h one Dee  only.  These problems seemed t o d i s a p p e a r when a two Dee r e s o n a t o r was used.  However, such p r e c a u t i o n s as c l e a n l i n e s s o f the r e s o n a t o r p a n e l s , p u l s i n g o f the RF a m p l i f i e r and a good p r e s s u r e o f the o r d e r 10 p a c t o r i n g was to be overcome. conditioning.  T o r r were mandatory i f m u l t i -  Sparks u s u a l l y o c c u r r e d d u r i n g a p e r i o d o f  A f t e r c o n d i t i o n i n g i t was p o s s i b l e to h o l d a s t a b l e Dee-to-Dee g  v o l t a g e of 200 kV f o r s e v e r a l hours,  p r o v i d e d the p r e s s u r e was l e s s than 5x10  Torr. A s i n g l e c o u p l i n g loop e x c i t e d s u c c e s s f u l l y a two Dee r e s o n a t o r . The ceramic vacuum feedthrough  f a i l e d due t o mechanical  r e p l a c e d by a t e f l o n s e a l . A resonant  Design  s t r e s s e s and had to be t e m p o r a r i l y  o f a new ceramic  feedthrough  t r a n s m i s s i o n l i n e performed s a t i s f a c t o r i l y .  i s i n progress.  A d e s i r e d impedance  match c o u l d e a s i l y be a t t a i n e d by making use o f two c a p a c i t o r s connected the l i n e .  However, s i n c e the r e s o n a t o r frequency was n o t s t a b l e (±.01%), one  c o u l d o n l y say t h a t an impedance match was o b t a i n e d at a frequency the r e s o n a t o r frequency different  along  ( w i t h i n 5 kHz).  from the r e s o n a t o r frequency  very c l o s e to  Resonant o p e r a t i o n a t a frequency (Af > 1 kHz) would l e a d t o h i g h e r l o s s e s  i n t h e c o u p l i n g loop assembly and, t h e r e f o r e , should be avoided. The  e l e c t r i c a l c h a r a c t e r i s t i c s o f t h e r e s o n a t o r w i t h t h e c e n t r e post and  cut-out included are p r e s e n t l y being v e r i f i e d  (CR c y c l o t r o n ) .  - 137 -  REFERENCES 1.  E.W. Vogt and J . J . B u r g e r j o n , e d i t o r s , "TRIUMF P r o p o s a l and Cost E s t i m a t e " (1966)  2.  E.G. A u l d e t a l . , "Design of the 4000 Ton Magnet f o r t h e TRIUMF C y c l o t r o n " i n P r o c e e d i n g s , I n t e r n a t i o n a l C y c l o t r o n Conference, 5 t h , Oxford, 1969 ( B u t t e r w o r t h s , London)  3.  L. Root and E.W.  4.  L.P. Robertson e t a l . , " E x t r a c t i o n of M u l t i p l e Beams o f V a r i o u s E n e r g i e s from the TRIUMF Negative Ion Isochronous C y c l o t r o n " i n P r o c e e d i n g s , I n t e r n a t i o n a l C y c l o t r o n Conference, 5 t h , Oxford, 1969 ( B u t t e r w o r t h s , London,1971)  5.  J.R. R i c h a r d s o n and M.K. Craddock, "Beam Q u a l i t y and Expected Energy R e s o l u t i o n f o r the TRIUMF C y c l o t r o n " i n P r o c e e d i n g s , I n t e r n a t i o n a l C y c l o t r o n Conference, 5 t h , Oxford, 1969 ( B u t t e r w o r t h s , London, 1971)  6.  K.L. Erdman e t a l . , "A 'Square Wave' RF System Design f o r the TRIUMF C y c l o t r o n " i n P r o c e e d i n g s , I n t e r n a t i o n a l C y c l o t r o n Conference, 5 t h , Oxford, 1969 ( B u t t e r w o r t h s , London, 1971)  7.  G.M. S t i n s o n e t a l . , " E l e c t r i c D i s s o c i a t i o n o f H F i e l d s " , TRI-69-1 (1969)  8.  E.W.  9.  J.R. R e i t z and F . J . M i l f o r d , Foundations of E l e c t r o m a g n e t i c Theory (Addison-Wesley, Reading, Mass., 1967)  Blackmore,  p r i v a t e communication  (1970)  Ions by Magnetic  Blackmore and G. Dutto, p r i v a t e communication  (1971)  10.  0. Dambach, p r i v a t e communication  (1971)  11.  R. Gummer, p r i v a t e communication  12.  J.R. R i c h a r d s o n , p r i v a t e communication  (1969)  13.  J.R. R i c h a r d s o n , p r i v a t e communication  (1969)  14.  W.D. K i l p a t r i c k , "A C r i t e r i o n f o r Vacuum S p a r k i n g Designed both RF and DC", Report U.C.R.L.-2321 (1953)  15.  B.H. Smith, "Radio-frequency System o f the B e r k e l e y 88 Inch C y c l o t r o n " , N u c l . I n s t r . & Meth. 18 (1962)  16.  J.C. S l a t e r , Microwave T r a n s m i s s i o n (McGraw-Hill, New York, 1942)  17.  Reference Data f o r Radio E n g i n e e r s , 5 t h e d i t i o n I n d i a n o p o l i s , 1969)  18.  P r o p o s a l f o r a 150 kW A m p l i f i e r , a 1.5 MW RF A m p l i f i e r and a 2.4 MW Power Supply, C o n t i n e n t a l E l e c t r o n i c s M a n u f a c t u r i n g Co., D a l l a s , Texas (1971)  (1971)  to Include  (Howard W. Sams,  R.F COUPLING LOOPS  iOOkv/  FLUX COUPLINGS  Particle Frequency- 4.62 R.F Frequency•• 23  4>O  R.F SKIN LOSSES TOTAL R.F POWER + IOOkV + IQOkV 200kV Fig. 3  Resonator system  synthesizer remote auto/synthl local _ auto/synth  freq.  RF sample input selec.  manual RF ' input level — -?=1 mod.  meter  phase det.  ohase shift  safety control  fundamental  IF a m p l i f i e r  3rd *** harmonic  trans, line  fundi >  root current e s o n a t o r loop  current  J  I  tuning plunger driver  screen -s> supply  frequency ranslato screen operating point  set 3rd harm. amplitude  3rd harmonic 3rd harm, input '"amplifier  Radio-frequency system  3rd harmonic r e l . amplitude: kphase mate  loop  loop  amp.  amp. ...  i-O  pnase e r r o r to tripler  input  screen 5  -  trans. line  mod.  Fig.  phase det.  operating  set point  3rd harmoni phase  - 143 RF fundamental v o l t a g e amplitude V (kV)  80  1600  90  100  110  120  S 1400 u CO  1200  IOOO L  800 L  600  L  400  1400  3 to  CO  o  u  1200  L  IOOO  r —  o PH  800  L  600 c h a r a c t e r i s t i c impedance o f segment Z (ft) Power l o s s i n two Dees a) as a f u n c t i o n o f s p e c i f i c  c o n d u c t i v i t y and  RF fundamental v o l t a g e amplitude b) as a f u n c t i o n o f c h a r a c t e r i s t i c impedance and r e s o n a t o r fundamental f r e q u e n c y  o U4  ^4000  2000  L  2000  4 specific  Fig.  7  5 conductivity  6  7  8  a (Mhos/m)  Resonator q u a l i t y f a c t o r a) vs c h a r a c t e r i s t i c impedance o f r e s o n a t o r segment Z b) vs s p e c i f i c c o n d u c t i v i t y o and r e s o n a t o r fundamental frequency f Q  Fig. 8  Resonator hot arm d e f l e c t i o n a) r e s o n a t o r made up o f n b) l i n e a r a p p r o x i m a t i o n c) p a r a b o l i c a p p r o x i m a t i o n  sections  t i p d e f l e c t i o n Ag Fig. 9  Percentage  (mm)  f r e q u e n c y change vs hot arm  deflection  - 147 -  M d) -o-  =F C R  Fig.  10  L  -AAAA/V-  Ri  6  Lumped parameter r e p r e s e n t a t i o n of the system a) system c o n s i s t s o f a s i n g l e r e s o n a t o r b) system c o n s i s t s of a one Dee r e s o n a t o r and a resonant line, c) system c o n s i s t s of a two Dee r e s o n a t o r and a resonant l i n e d) e q u i v a l e n t r e p r e s e n t a t i o n f o r a)  150  15  100  10  > ^4  0)  -a  3  a;  60 CO  o CO  c o  f i r s t harmonic, f = 22.66 MHz, two Dee r e s o n a t o r and r e s o n a n t l i n e  CO  ai  t h i r d harmonic, f  = 67.98 MHz. two Dee °3 r e s o n a t o r and resonant l i n e f i r s t harmonic f = 2 2 . 6 6 MHz, one Dee oi resonator f i r s t harmonic, f = 2 2 . 6 6 MHz, one Dee r e s o n a t o r and resonant l i n e  50  100 t  Fig.  11  (usee)  Resonator v o l t a g e amplitude  r i s e during  150 transient  200  0  -  r  — 03  >  149  -  one Dee  resonator  two  Dee  r e s o n a t o r and  resonant  l i n e , Main  two  Dee  r e s o n a t o r and  resonant  line,  one Dee two Dee  r e s o n a t o r and resonant r e s o n a t o r and resonant  CR  cyclotron  l i n e , CR l i n e , CR  cyclotron cyclotron  5.0  CO  •x) =»  4->  •r-l .-I  i*  CC  cu  00 GJ 4->  1  2.5  u o AJ a c o w cu u  0.0  25  0.0  50 t  Fig.  12  Initial  cyclotron  (usee)  r i s e of r e s o n a t o r v o l t a g e  amplitude  4  hot arm  F i g . 13  Resonator  length I  (cm)  resonant frequency vs h o t arm  length  F i g . 14  Resonator resonant frequency vs t i p l o a d i n g  capacity  - 153 -  b)  —4 1 o  i  i  s.  4-  >  e gap Ag  F i g . 16  s  a  /o  (mm)  P e r c e n t a g e f r e q u e n c y change vs p o s i t i o n o f c a p a c i t i v e a) the fundamental b) the t h i r d harmonic  plates  - 154  a)  parabolic  -  approximation  measured v a l u e s linear  approximation  f , = 23.10 ol  - 6  -4  MHz  0  -2  t i p d e f l e c t i o n Ag 1  1  _  parabolic  (mm)  1  approximation  measured v a l u e s linear  approximation  f , = 69.27 o3  6  _4  6 t i p d e f l e c t i o n Ag  F i g . 17  MHz  (mm)  Resonator resonant frequency vs ground arm t i p d e f l e c t i o n (CRM) a) the fundamental b) the third, harmonic  - 155  a)  -I  6^ C 6D 0  -2  is  Xi  o  computed v a l u e s , f  CJ  C  cu 3  -3 ~X  cr cu u  measured v a l u e s , f 'measured v a l u e s , f  -4 •  ol  0 3  computed v a l u e s , f  0 3  139.52 MHz 139.52 MHz  -5 6  8  10  t i p d e f l e c t i o n Ag  12  14  16  (mm)  b)  IOOOO  T  —o  o  8G00  measured v a l u e s ,  f i r s t harmonic  measured v a l u e s ,  t h i r d harmonic  u  o o ca  w  6000  u  X -o-  rH H CO  cr  X  _X_  X  _o_  -x-  -X.  -o-  x -o  4000  2000  8  10  t i p d e f l e c t i o n Ag (mm) F i g . 18  14  P e r c e n t a g e frequency change a) and q u a l i t y f a c t o r v a r i a t i o n b) vs ground arm t i p d e f l e c t i o n ( h a l f - s c a l e r e s o n a t o r ) .  16  - 156 -  F i g . 19  Schematic o f the t u n i n g b e l l o w s  - 157 -  Ll  F i g . 20  (mm)  P e r c e n t a g e f r e q u e n c y change vs p o s i t i o n of t u n i n g b e l l o w s (CPJ1) a) the fundamental b) the t h i r d harmonic; c a l c u l a t i o n based on p e r t u r b a t i o n t h e o r y by S l a t e r .  Fig.  21  Percentage frequency change vs p o s i t i o n o f diaphragm i n the r e s o n a t o r c a l c u l a t e d u s i n g p e r t u r b a t i o n theory by S l a t e r (zero p o s i t i o n r e f e r s to the r o o t ; the diaphragm i s i n i t s v e r t i c a l p o s i t i o n )  p o s i t i o n o f diaphragm  F  i  8  '  2  2  (cm)  F i g . 23  Extreme end s e c t i o n a) top view b) s e c t i o n w i t h m o d i f i e d hot  24  - C e n t r a l r e s o n a t o r segments a) top view b) s i d e view c) g r i d o f p o i n t s f o r v o l t a g e measurements  25  Beam-RF f i e l d i n t e r a c t i o n a) r e s o n a t o r f e d by an RF g e n e r a t o r and a beam c u r r e n t g e n e r a t o r b) amplitude o f a beam p u l s e c ) r e p r e s e n t a t i o n of a r e s o n a t o r w i t h a beam l o a d d) beam induced v o l t a g e and t o t a l v o l t a g e on r e s o n a t o r  F i g . 26  Power d e l i v e r e d to the beam a) and frequency change b) v s i n j e c t i o n phase fundamental, I = 100 uA, E = 500 MeV)  (the RF  f f  = 22.66 MHz 03  one Dee resonator  = 67.98 MHz  q + p = 80 deg  b)  500  N M  250  < CJ 60  rt •d u  o c o* u CJ  -250  L_  90  - 60  I  - 30  0  pulse p o s i t i o n  F i g . 27  -500  L  30  60  (degrees)  90  JL  90  -60  _L  -30  0  pulse p o s i t i o n  30 (degrees)  Power delivered to the beam a) and frequency change b) vs i n j e c t i o n phase (the t h i r d harmonic, I = 100 uA, E = 500 MeV)  60  90  F i g . 28  Power d e l i v e r e d to the beam a) and frequency change b) vs i n j e c t i o n phase fundamental, I = 750 uA, E = 400 MeV)  (the RF  f o l, = 22.66 MHz f  03  - 67.98 MHz  one Dee r e s o n a t o r  q + p = 80 deg a)  b)  10  2.5  N <4-( <  QJ 60  C  ca  * 0  x  CJ  o c  H  u  u  0.0  cu  3 cr CD  w  CH  -10 -90  -60  2.5  -30  -90  pulse p o s i t i o n  F i g . 29  (degrees)  -60  -30  30  0  pulse p o s i t i o n  (degrees)  Power d e l i v e r e d to the beam a) and frequency change b) v s i n j e c t i o n phase harmonic, I = 750 uA, E = 400 MeV) B  60  (the t h i r d  90  f f  ol 03  = 22.66 MHz = 6 7 . 9 8 MHz  one Dee r e s o n a t o r  a)  b)  30.0  0.0 •  1  '  f-27.5  P-I  CD  25  ^  1  1  ~—  _  CU  5-1  u  c e n t r e o f p u l s e s e t at 0 deg o f RF phase c e n t r e o f p u l s e s e t at -20 deg o f RF phase  - 5.0  25.0 20  40  beam phase w i d t h  Fig.  30  60 (degrees)  80  100  1  1  l  20  40  60  beam phase w i d t h  •  80  100  (degrees)  Power d e l i v e r e d to the beam vs beam phase w i d t h a) t h e RF fundamental b) the t h i r d (I = 100 uA, E = 500 MeV)  harmonic  I  frequency change Af (Hz) ro  04  o  i  ro  o  OQ  t—' CT w  ft) cu  3  XS IT)  l-t ft)  rt ,n  CD tt> 3  rt O H. O a. 3" cu 3* 3 TO t-i 3 fD So  o 3 cn HO  3" Cu CO  ft) Cu  rt . 3* o  Cu ft) TO i-l ft) ft)  KJ  —I  CD  ON  00  to  cy.  CO  3  XS  cu CO  o  tt> S3  H>  3  frequency change Af (Hz)  o O  ft>  O O  o  a  to ft>  n  fD CO  o 3  cu rt o  O  o  1-1  •c  >  o o  s It)  <  3* CD Hi  3 3  Cu  tt) 3 rt Cu  - 89T -  Fig.  32  T o t a l RF fundamental v o l t a g e on r e s o n a t o r d u r i n g the i n j e c t i o n o f a beam (without the RF t h i r d harmonic)  Fig.  33  T o t a l RF fundamental v o l t a g e on r e s o n a t o r d u r i n g the i n j e c t i o n o f a beam (with, the RF t h i r d harmonic)  12  ;  E  g  —Eg V  o l  - 500 MeV,  I  = 400 MeV,  I  - 113 kV, V  g  o 3  = 100  yA  = 750  yA  = 13 kV  -p = -36 deg q =  0  100  200  36 deg  300  t - t . (ysec)  Fig.  34  T o t a l RF t h i r d harmonic v o l t a g e on r e s o n a t o r (with the RF t h i r d harmonic)  during  the i n j e c t i o n o f a beam  400  - 172 -  RF fundamental v o l t a g e wave f l a t - t o p p e d v o l t a g e wave  \  \  \  —  —  A  RF t h i r d harmonic v o l t a g e wave  /  /  w----7  \ \  / V  211  V  \  \  /  \  \  -p = - 52 deg, q =-28 deg -p = - 12 deg, q = 12 deg i_„  o  •  II  :  .  2ir  CO t  35  Components o f r e s o n a t o r v o l t a g e d u r i n g the course o f a c c e l e r a t i o n a) RF v o l t a g e amplitudes due t o e x t e r n a l sources b) the f i r s t harmonic component of beam induced v o l t a g e (RF fundamental o p e r a t i o n )  p = -36 deg, q = 36 deg  Fig.  36  Components o f r e s o n a t o r v o l t a g e d u r i n g the course o f a c c e l e r a t i o n (RF f l a t - t o p o p e r a t i o n ) a) the f i r s t harmonic component o f beam induced v o l t a g e b) the t h i r d harmonic component o f beam induced v o l t a g e  - 175 -  a) RL  B  -AA/V  v:  v,  B o-  •o  F i g . 38  R e p r e s e n t a t i o n o f c o u p l i n g a) and lumped parameter r e p r e s e n t a t i o n o f a Dee-resonant l i n e system b)  measured v a l u e s , f  a)  _  computed v a l u e s  22.937 MHz ol (lumped parameter r e p r e s e n t a t i o n ) ,  computed v a l u e s  (program MATCH),  f  f  = Q l  22.660 MHz  22.660 MHz  1000  750 eg CW  O  CU  500 _ cfl S3  250  -5.0  -Z5  0  2.5  frequency change Af (kHz) F i g . 39  5.0  -5.0  -2.5  0  2.5  frequency change Af (kHz)  Input impedance vs d e t u n i n g from resonance a) magnitude o f i n p u t impedance b) phase o f i n p u t impedance  5-0  - 177 -  150  2.5  0.0 Af  2-5  (kHz)  5-0  measured v a l u e s -  F i g . 40  Resonator  computed v a l u e s  (lumped parameter r e p r e s e n t a t i o n )  peak v o l t a g e vs d e t u n i n g from  resonance  "A" DEE Tn  "B"DEE  lite l a .  B Te '7  DETAIL A  A  T  6  T  T  4 z  T  7  y  c  T I  2  5  L-L T  5  DETAIL C o  H  Co  E-E  DETAIL B  "C  3 /  L A C D  F i g . 41  Representation  = = = =  .D  T  i  length height thickness width  o f the system w i t h d i s t r i b u t e d  parameters  (program RESLINE)  6  <  R  TUBE  - 179 -  JOLL  A -o-  B  nA/2 'TUBE  •o  B' XL+X  A o-  TL  ,  ±  B  Xc  V,  A' B'  Matching u s i n g a nA/2 l i n e a) r e p r e s e n t a t i o n o f a Dee w i t h lumped parameters and a matching network b) Dee and l i n e i n the v i c i n i t y of t c) impedance t r a n s f o r m a t i o n u s i n g IT network Q  Fig.  43  R e p r e s e n t a t i o n of the system a) two Dees r e p r e s e n t e d w i t h lumped parameters and the t r a n s m i s s i o n l i n e w i t h d i s t r i b u t e d parameters (program MATCH) b) lumped parameter r e p r e s e n t a t i o n o f the whole system  - 181 -  f  o  = 23.1  0  -10 driving F i g . 44  MHz  frequency  10  change Af (kHz)  Phase, s h i f t between l o o p and r e s o n a t o r r o o t c u r r e n t s v s frequency  driving  - Z8T -  - 183 a)  -4  0  4  stub s h o r t i n g  12  8  16  p l u n g e r p o s i t i o n Ad (cm)  b)  OB f K  0.6  a  oi  = 47.61 MHz  , - x/4  stub  o Ad <* A/4 - I  <] cu oo  I  stub  0.4  u  c  0) S* Q2 M-l  0.0  1  4 stub s h o r t i n g  46  8  12  plunger p o s i t i o n  16 Ad (cm)  P e r c e n t a g e f r e q u e n c y change vs stub s h o r t i n g p l u n g e r p o s i t i o n a) 4 s t u b s coupled c a p a c i t i v e l y to a Dee made up o f 10 s e c t i o n s b) 4 s t u b s coupled c a p a c i t i v e l y t o a two Dee r e s o n a t o r , each Dee made up o f 10 s e c t i o n s  F i g . 47  V o l t a g e v a r i a t i o n along the a c c e l e r a t i n g gap vs stub s h o r t i n g p l u n g e r p o s i t i o n a) 4 stubs coupled c a p a c i t i v e l y to a Dee made up o f 10 s e c t i o n s b) 4 stubs c o u p l e d c a p a c i t i v e l y t o a two Dee r e s o n a t o r , each Dee made up o f 10 s e c t i o n s  T  F i g . 48  1  1  r  Percentage frequency change and q u a l i t y f a c t o r v a r i a t i o n vs stub s h o r t i n g p o s i t i o n (a stub coupled c a p a c i t i v e l y to a Dee made up o f 5 s e c t i o n s )  plunger  - 186 -  F i g . 49  V o l t a g e v a r i a t i o n along the hot arm t i p s vs ground arm t i p d e f l e c t i o n (Dee made up o f 5 s e c t i o n s ) a) the fundamental b) the t h i r d harmonic  - 187 -  a)  0.0  < -0-2 cu CO  c n)  J2 o  >. - 0 . 4  o c  OJ  3  f  cr a)  u  f  -0.6  ol  = 46.337 MHz  „ = 139.608 MHz o3  -0.8 0.0  0.5  1.0  1.5  p o s i t i o n of capacitors  b)  5000  T  ±  ±  2.0  2.5  (cm)  T  T ©  4000  -  S 3000  -•  f i r s t harmonic Q  -A  t h i r d harmonic Q  CJ  ca  3 2000 3  0*  1000  0 0.0  0.5  1.0  1.5  p o s i t i o n of c a p a c i t o r s  Fig.  50  2.0  2.5  (cm)  Frequency t u n i n g by means o f c y l i n d r i c a l c a p a c i t o r s (Dee made up o f 10 s e c t i o n s ) a) p e r c e n t a g e frequency change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n o f c y l i n d r i c a l c a p a c i t o r s  -  Fig.  51  188  -  V o l t a g e v a r i a t i o n a l o n g the hot arm c y l i n d r i c a l capacitors  t i p s vs p o s i t i o n  of  189 -  5000  a)  :—i  T  1 —  — f i r s t  4000  A—  A  harmonic Q  t h i r d harmonic Q  u o o  4->  £ 3000 4-1  3  2000  1000  f  ol  = 45.2 MHz  f = 137.4 MHz - o3  _L  QO  0.2  0.4  0.6  0.8  gap Ag (cm)  b)  > <  0 60)  f.  = 45.2 MHz  ct)  Xi o  C 6«U 0 0 < 4-1  X-  A-  -x  *  x  x  x  -A  A  A  A  A  A.  ^  rH O >  -<-»—•  >  -2  0.0  -A  A"  0.2  ±  0.4  A~  -  "A  o-  5~ ~ A  J_  0.6  0.8  gap Ag (cm) Fig.  52  Q u a l i t y f a c t o r v a r i a t i o n a) and v o l t a g e v a r i a t i o n a l o n g the h o t arm t i p s b) v s p o s i t i o n o f c a p a c i t i v e t u n i n g p l a t e s (Dee made up of 10 s e c t i o n s )  a)  b)  p o s i t i o n of f i n s  F i g . 53  (degrees)  position  of f i n s  (degrees)  Frequency t u n i n g by means of r o t a t i n g f i n s (measurements done w i t h a 3 s e c t i o n r e s o n a t o r w i t h t h e f i n s i n s e r t e d i n the upper and lower c e n t r e segments) a ) p e r c e n t a g e f r e q u e n c y change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n o f t u n i n g f i n s  Fig.  54  Frequency t u n i n g by means o f r o t a t i n g f i n s (measurements done w i t h a two s e c t i o n r e s o n a t o r w i t h 8 f i n s p e r each segment) a) percentage frequency change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n of f i n s  b)  a)  2.0  10000 f  ol  1—  _ t h i r d harmonic Q  = 47.44 MHz  f , = 143.17 MHz  -A  o3  Rl.5  '  1  &  f i r s t harmonic Q  7500  -  MH  <  u  o •w o  60  c (0  4H  o >.  5000  .0  •H  U  C 0) 3  3  cr  cr  CU M  2500  0.5  0.0  I  30 p o s i t i o n of loops  F i g . 55  60 (degrees)  90  30 p o s i t i o n of loops  60  90  (degrees)  Frequency t u n i n g by means o f r o t a t i n g l o o p s (measurements done w i t h a two s e c t i o n r e s o n a t o r w i t h 8 l o o p s p e r each segment) a) percentage frequency change and b) q u a l i t y f a c t o r v a r i a t i o n vs p o s i t i o n of loops  - 19.3 -  a)  Resonator T I I U  Power amplifier  Tripler  Digital voltmeter  4 — TInTI" — V  TE  LJ  4-  4"  4  4  JTA  Harmonic analyzer  vector voltmeter  Frequency counter  C.R.T.  Oscillator  b)  Fig.  56  Resonator f r e q u e n c y t u n i n g by means o f diaphragms a) schematic o f the e x p e r i m e n t a l arrangement b) r e s o n a t o r w i t h t u n i n g diaphragms  - 19,4 -  a)  0.0  -0.5  <u 60  c  CD X! O  -1.0  o c 3 a*  0)  <u M  -1.5  -f  -2.0  _J  0  ol  30  = 22.6508 I  MHz  I  60  90  p o s i t i o n o f diaphragms (degrees)  b)  3.02  3.00 M O  a cu 3  cr CD S-1  2.98 -  2.96  30  60  90  p o s i t i o n o f diaphragms (degrees)  Fig.  57  Frequency t u n i n g by means o f t u n i n g diaphragms a) p e r c e n t a g e f r e q u e n c y change and b) t h i r d - t o - f i r s t harmonic frequency r a t i o vs p o s i t i o n of diaphragms; p o s i t i o n of diaphragms i n r e s o n a t o r at 108.5 i n . from r o o t  - 195 -  a)  ,-0.5 <  CU §  -1.0  o >>  o c  CU  3  •f  cr  c u u  m  -f  15  f f  - A -  -2.d 0  ol o3  . 21.65 i n . from r o o t . 65.4 i n . from  , 6 5 . 4 i n . from r o o t oi' o3  , 2 1 . 6 5 i n . from r o o t  _L_  30  60  p o s i t i o n o f diaphragms b)  root  3.02  T  T  3.00  f  03  90 (degrees)  I  "x  x  65.4 i n . from  root  -e  © — 21.65in. from root-  = 67.894 MHz  2.94 0  30  60  p o s i t i o n of diaphragms Fig'. 58  90 (degrees)  Frequency t u n i n g by means o f t u n i n g diaphragms a) p e r c e n t a g e f r e q u e n c y change and b) t h i r d - t o - f i x s t harmonic frequency r a t i o v s p o s i t i o n o f diaphragms; p o s i t i o n o f diaphragms i n r e s o n a t o r a t 65.4 i n . and 21.65 i n . from r o o t  - 19.6 -  Fig.  59  Resonator frequency t u n i n g by means o f a t u n i n g stub a) e x p e r i m e n t a l arrangement b) lumped c o n s t a n t r e p r e s e n t a t i o n  F i g . 60  P e r c e n t a g e v o l t a g e change vs h o t arm d e f l e c t i o n a) d e f l e c t i o n o f the upper h o t arm #3 i n an 18 . s e c t i o n one Dee r e s o n a t o r b) d e f l e c t i o n of the lower h o t arm #6B i n a 20 s e c t i o n two Dee r e s o n a t o r  b)  -  4  -  2  0  number of t u r n s  Fig.61  Percentage 18 s e c t i o n  2  (0.05 i n . / t u r n )  4  -4  -2 number o f t u r n s  0  2  4  (0.05 i n . / t u r n )  frequency change vs h o t arm d e f l e c t i o n a) d e f l e c t i o n o f the upper h o t arm #3 i n an one Dee r e s o n a t o r b) d e f l e c t i o n o f the lower h o t arm #6B i n a 20 s e c t i o n two Dee r e s  F i g . 62  Percentage voltage 18 s e c t i o n one Dee Dee r e s o n a t o r  change vs r o o t p l u n g e r p o s i t i o n a) motion o f r o o t p l u n g e r s i n s e c t i o n #7 r e s o n a t o r b) motion of r o o t p l u n g e r s i n s e c t i o n #7A i n a 20 s e c t i o n two  in  an  - 200 -  a)  r—  0.2  U-l <  0.1  01 00  o  0.0  o c  <D  cr  0)  u  0.1 f  oi  = 47.781  MHz  -0.2 -4  -2  0  2  p o s i t i o n o f r o o t p l u n g e r s Aft  4 (cm)  b)  0.2  -2  0  2  p o s i t i o n o f r o o t p l u n g e r s Aft  F i g . 63  4 (cm)  P e r c e n t a g e frequency change vs r o o t p l u n g e r p o s i t i o n a) motion of r o o t p l u n g e r s i n s e c t i o n #7 i n an 18 s e c t i o n one Dee r e s o n a t o r b) b) motion o f r o o t p l u n g e r s i n s e c t i o n #7A i n a 20 s e c t i o n two Dee resonator  - 201  a)  0.5  -I  0  position  of root  I  2  p l u n g e r s A£ (cm)  b)  5000  4000 u o u  3000  2000  000  -S—  f i r s t harmonic Q  A —  t h i r d harmonic Q  -L  -3  -2  -I position  F i g . 64  0 of r o o t  I  2  p l u n g e r s A£ (cm)  Percentage f r e q u e n c y change a) and q u a l i t y f a c t o r v a r i a t i o n b) vs motion o f r o o t p l u n g e r s i n s e c t i o n #4 i n a 5 s e c t i o n one Dee r e s o n a t o r  //// ///  I'll  ////  III  III IP II  EQUIPOTENTIALs\\\  In  II  ('  ll ill  A  U  F i g . 65  a.  C e n t r a l r e g i o n geometry  300  T  1  T  measured v a l u e s c a p a c i t o r CP(3) = 400 pF computed v a l u e s p o s i t i o n o f CP(3): 342 cm c a p a c i t o r CP(2) = 100 pF  200  c a p a c i t o r CP(6) = 87 pF  POiNT OF REFERENCE 0)  oo rt o >  100  -  16 LOOP  F i g . 66  24  26  28  29  number o f s t a t i o n  V o l t a g e d i s t r i b u t i o n a l o n g the t r a n s m i s s i o n l i n e ; b a s i c  30  31  32  33 TUBE  set—up  300  I  1  T  T  -measured v a l u e s c a p a c i t o r CP(3) = 415 pF •computed v a l u e s p o s i t i o n o f CP(3) » 342 cm c a p a c i t o r CP(2) = 100 pF  200  POINT OF R E F E R E N C E  c a p a c i t o r CP(6) = 70 pF = 2345 ft  OJ  4-1  100  32 LOOP  Fig.  67  number o f s t a t i o n  V o l t a g e d i s t r i b u t i o n along the transmission; l i n e ; magnitude o f CP(3) changed t o 415 pF  33 TUBE  300  -9-  -measured v a l u e s c a p a c i t o r CP(3) = 400 pF ..computed v a l u e s p o s i t i o n o f CP(3) = 3 2 7 cm c a p a c i t o r CP(2) = 100 pF  200  c a p a c i t o r CP(6) = 74 pF  POINT OF REFERENCE  100  0  16 LOOP  F i g . 68  18  24  26  28  29  number o f s t a t i o n  V o l t a g e d i s t r i b u t i o n a l o n g the t r a n s m i s s i o n l i n e ; p o s i t i o n  30  31  32  33 TUBE  o f CP(3) changed t o 327 cm  360  I  I-  -measured values ^computed values  270  0)  oo rt  180  4-1  r H  o  capacitor CP (3) •> 400 pF position of CP(3): 342 cm capacitor CP(2) = 100 pF  90 L  capacitor CP(6) = 87 pF UBE  ^ P O I N T  O  F  R  E  F  E  R  E  N  C  E  CP(2)  16 LOOP  Fig. 69  18  = 2000 Q  20  22  24  CP (3) »  26  CP (6)  28  29  number of station Voltage phase along the transmission l i n e ; basic s e t -up  30  31  32  33 TUBE  f-90U  C  fO  O  oo  F i g . 71  RF c o n t a c t s  - 209 -  Appendix A:  CENTRE REGION CYCLOTRON RF SYSTEM PARAMETERS - THE RF FUNDAMENTAL  - 210 20:04:02 TRAP TRITUME 3 DATE 01-05-72 CENTRE REGION 8 SECTIONS I N 2 DEES ^RESONATORS* 23.1000 MHZ RESONANT FREQUENCY 38.30 OHMS CHARACTERISTIC IMPEDANCE 7. 0 0 P FARAD T I P TO T I P C A P A C I T A N C E VOLTAGE,CURRENT PEAKS,POWER L O S S RMS AT S H O R T X = 0 V O L T A G E TO C U R R E N T P H A S E 9 0 D E G R E E S 3.08622 METERS HOT A R M L E N G T H 0. 1 5 2 4 METERS T I P TO T I P D I S T A N C E 1.0000 METERS A V E R A G E W I D T H OF S E C T I O N 99998.4 VOLTS MAX.VOLTAGE ON R E S # 1 100004.0 VOLTS M A X . V O L T A G E ON R E S # 2 179.84 DEGREES VOLTAGE PHASE SHIFT 2 6 1 8 .8 AMPS MAX.CURRENT IN RES#1 2619.0 MAX.CURRENT IN RES#2 AMPS 26. 188 C U R R E N T D E N S I T Y AT ROOT AMPS/CM 5000. POWER L O S S D U E T O B E A M WATTS POWER L O S S I N R E S O N A T O R S 120313. WATTS POWER L O S S I N R E S # 1 WATTS 57479. POWER L O S S I N R E S # 2 57834. WATTS ^COUPLING LOOP* VOLTAGE INDUCED I N LOOP 9277.2 VOLTS LOOP D I M E N S I O N S ARE HIGHT = 1.75 INCHES LENGTH = 17 . 5 0 INCHES WIDTH = 3.00 INCHES THICKNESS = 0.50 INCHES 3.5 0 LOOP POSITION INCHES 0.224169 LOOP S E L F I N D U C T A N C E MICROHENRY 25.94 CURRENT THROUGH LOOP AMPS 351.9 POWER L O S S I N L O O P ' WATTS *TRANSMISSION L I N E * C H A R . I M P E D A N C E OF L I N E = 49.78 OHMS OUTER DIAMETER = 11.750 INCHES INNER DIAMETER = 5.125 INCHES MAX.VOLTAGE WITHOUT C P ( 3 ) = 15531. VOLTS MAX.CURRENT WITHOUT C P ( 3 ) = 312. AMPS MAX.VOLTAGE WITH C P ( 3 ) = 22802. VOLTS MAX.CURRENT WITH CP(3) = 458. AMPS CU RRENT THROUGH C P ( 2 ) = 186.62 AMPS C A P A C I T O R A F T E R LOOP C P ( 2 ) = 150.000 PFARAD P O S I T I O N OF C P ( 2 ) = 0.867 METERS CURRENT THROUGH CP(3) = 600. AMPS CAPACITOR C P ( 3 ) = 355.000 PFARAD P O S I T I O N OF CP(3) = 3.8200 METERS LENGTH AFTER C P ( 3 ) '= 2.5500 METERS METERS T O T A L L E N G T H OF L I N E = 6.3700 VSWR -WITHOUT C P { 2 ) = 7.25 VSWR W I T H O U T C P ( 3 ) = 20.12 VSWR A F T E R C P ( 3 ) = 42.93 POWER L O S S I N L I N E = 1563. WATTS POWER L O S S I N N E R C . = 1088. WATTS POWER L O S S O U T E R C . = 475. WATTS *POWER T U B E * TUBE CURRENT 16 . 6 2 AMPS TUBE VOLTAGE 14669.9 VOLTS C U R R E N T TO V O L T A G E PHASE 0.00 DEGREES CAPACITOR C P ( 6 ) AFTER LINE = 164.581 PFARAO CURRENT THROUGH C P ( 6 ) 350.4 AMPS T O T A L POWER L O S S 121879. WATTS  ^RESONATORS* AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 1 AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 2 Q U A L I T Y COMPUTED QUALITY MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA  1 1  = =  2 2  = =  FORESHORTENING CONDUCTIVITY SKIN DEPTH A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l = L U M P E D I N D U C T A N C E AT V O l LUMPED C A P A C I T A N C E AT V 0 2 = L U M P E D I N D U C T A N C E AT V 0 2 LUMPED C A P A C I T A N C E AT 101 = LUMPED I N D U C T A N C E AT 1 0 1 LUMPED C A P A C I T A N C E AT 102 = L U M P E D I N D U C T A N C E AT 1 0 2 EQUIVALENT VOLTAGE PEAK 1 = EQUIVALENT VOLTAGE PEAK 2 = ION O R B I T I N G F R E Q U E N C Y ACCELERATING TIME PERIOD TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON R E S O N A T O R POWER WHEN Q M E A S = SHUNT R E S I S T A N C E 1 AT V P E A K = SHUNT R E S I S T A N C E 2 AT V P E A K = SHUNT R E S I S T A N C E AT VLOOP= D E E TO D E E C A P A C I T A N C E AVERAGE ENERGY IN C P ( 1 ) T 0 T = R E A C T A N C E OF C T I P T O T A L D I S T R I B U T E D C A P A C I T Y DEE D I S T R I B U T E D INDUCTANCE DEE = RESISTANCE DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA I M A G . C O M P O N E N T OF Z R E S 2*L00P AREA/RESONATOR AREA RESONATOR STEPUP RATIO DEE S E R I E S R E S I S T A N C E P O W E R L O S S I N ROOT 1 POWER L O S S I N ROOT 2 M U T U A L I N D U C T A N C E AT LOOP C O U P L I N G C O E F F . A T LOOP MUTUAL INDUCTANCE D I S T R I B . M U T U A L I N D U C T A N C E AT T U B E COUPLING COEFF.AT TUBE  =  = = =  1.42109 1.42109 2.84218 1 .42125 1.42124 2.84249 7155.14 7 1 5 5 .14 7499.5 2 23.7160 12.98702 3.24675 145.14143 4.4501 5 .800 0.00137738 0.00003268 568.46 0.0835065 568.46 0.0835062 916.36 0.0518025 916.36 0.0518028 102150. 102142. 4.6200 1.52 98 . 6 0 35.0 120313. 86985.44 86461.38 357.6799 28.00 0.2800062 -246.07 348.129 0.0319167 0.0025035 0.1016 0.1016 0.39370 0.627119 0.4838048 -0.0025873 0.089112 10.77951 0.0016939 174 4.44 1744.63 0.0077468 0.05662043 0.006 207 2 0.0118390 0.00197989  JOULES JOULES JOULES JOULES JOULES JOULES  MHZ METERS METERS 10**6RAD/SEC DEGREES . 1 0 * * 7 MHOS CM 1/M PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY VOLTS VOLTS MHZ MICROSEC MICROSEC WATTS OHMS OHMS OHMS PFARAD JOULES OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS M ET ER S * * 2 1/M OHMS  OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY  - 212 •COUPLING LOOP* LOOP R E S I S T A N C E 0.0087623 LOOP REACTANCE 32.53622 LOOP A R E A 0.019758 INPUT CURRENT DENSITY 3.404 LOOP S U R F A C E 0.081290 0.0000377 A V E R A G E ENERGY I N LOOP 2.95 L O S S D U E TO I N P U T C U R R E N T = 348.93 L O S S D U E TO C A V I T Y C U R R E N T = •TRANSMISSION LINE* I M A G . C O M P O N E N T OF Z T L -0. 0045431 ALFA IN LINE 0.00004415 DISTRIBUTED CAPACITY 66.9569 DISTRIBUTED INDUCTANCE 0.165944 R E S I S T A N C E PER UNIT LENGTH = 0.004396 OUTER RADIUS 0.1492248 INNER RADIUS 0.0650874 AVERAGE E L E C T R I C ENERGY 0.0295545 AVERAGE MAGNETIC ENERGY 0.0295550 AVERAGE TOTAL ENERGY 0.0591095 TL Q U A L I T Y FACTOR 5489.83 LUMPED C A P A C I T A N C E AT V T U B E = 0.11086 LUMPED I N D U C T A N C E AT V T U B E = 428.17773 S E R I E S R E S I S T A N C E OF LINE = 11.3203 FIRST SECTION AVERAGE MAGNETIC ENERGY 0.0000877 AVERAGE ELECTRIC ENERGY 0.0011958 AVERAGE TOTAL ENERGY 0.0012835 POWER L O S S O U T E R C O N D U C T O R = 1.4 POWER L O S S I N N E R C O N D U C T O R = 3.1 POWER L O S S T O T A L 4.5 REFLECTION COEFFICIENT 0.757 R E A C T A N C E OF C P ( 2 ) -45.93 AVERAGE ENERGY IN C P ( 2 ) 0.0027554 SECOND S E C T I O N AVERAGE MAGNETIC ENERGY 0.0099556 AVERAGE ELECTRIC ENERGY 0.0019973 AVERAGE TOTAL ENERGY 0.0119529 POWER L O S S O U T E R C O N D U C T O R = 160. 1 POWER L O S S I N N E R C O N D U C T O R = 366.9 POWER L O S S T O T A L 527.0 REFLECTION COEFFICIENT 0.905 R E A C T A N C E OF C P ( 3 ) -19.41 1NIT.VOLTAGE MAX.POSITION = 5.317 THIRD SECTION AVERAGE MAGNETIC ENERGY 0.0194740 AVERAGE ELECTRIC ENERGY 0.0027316 AVERAGE TOTAL ENERGY 0.0222056 POWER L O S S O U T E R C O N D U C T O R = 313.2 POWER L O S S I N N E R C O N D U C T O R = 718.1 POWER L O S S T O T A L 1031.2 REFLECTION COEFFICIENT 0.954 P O S I T I O N OF V O L T A G E M A X I M U M = 8. 1 7 0 AVERAGE ENERGY IN C P ( 3 ) 0.0120197 *POWER T U B E * R E S I S T A N C E TO M A T C H 882.8618 R E A C T A N C E TO M A T C H 41.8629 REACTANCE OF C P ( 3 ) -41.8628 R E S I S T A N C E OF T U B E 882.8623 AVERAGE ENERGY IN C P ( 6 ) 0.00S8547  OHMS OHMS METERS**2 AMPS/CM METER**2 JOULES WATTS WATTS OHMS 1/M P FARAD/M MICROHEMRY/M OHMS/M METERS METERS JOULES JOULES JOULES PFARAD MICROHENRY OHMS JOULES JOULES JOULES WATTS WATTS WATTS OHMS JOULES JOULES JOULES JOULES WATTS WATTS WATTS OHMS METERS JOULES JOULES JOULES WATTS WATTS WATTS METERS JOULES OHMS OHMS OHMS OHMS JOULES  STN .METERS.. 0.0 1 0.6172 2 0.6172 3 0.6172 4 0.6172 5 0.6172 6 0.0 7 8 0.6172 9 0.6172 10 0.6172 11 0.6172 12 0.6172 13 0.0 14 0.0 15 0.0 16 0.0 17 0.0 18 0.1734 19 0.1734 20 0.1734 21 0.1734 22 0.1734 23 0.0 24 0.1969 25 0.1969 26 0.1969 27 0 . 1969 28 0.1969 29 0.1969 30 0.1969 31 0.1969 32 0.1969 33 0.196 9 34 0.1969 35 0 . 196 9 36 0.1969 37 0.1969 38 0.1969 39 0.0 40 0.1700 41 0.1700 42 0.1700 43 0.1700 44 0.1700 45 0.1700 46 0.1700 47 0.1700 48 0.1700 49 0.1700 50 0.1700 51 0.1700 52 0.1700 53 0.1700 54 0.1700 55 0.0 56 0.0 57 0.0  ,.E(1,K).., 0.0 29509.17 56406.30 78310.50 93283 .00 99998.44 -100003 .75 -93288.06 -78314.75 -56409.59 -29511 .05 -0.38 -25599.17 -4313.29 -92 77.21 -9275.16 -9275.16 -9242.26 -9144.36 -8982.14 -8756.74 -8469.75 -8469.75 -7198.78 -5862 .57. -4473.21 -3043 .30 -1585.80 -113.93 1358.98 2819.57 4254.60 5651 .07 6996.32 8278 .13 9484.92 10605 .72 11630.38 11630 .39 9979.79 8261.70 6487.78 4669.98 2820.61 952 .18 -922.70 -2791 .34 -4641.11 -6459.50 -8234.21 -9953.27 -11605.03 -13178 .33 -14662.56 -14662.56 -14662.56 -14662.56  - 213 .E(2,K) • ET ( K ) . •1 I ( 1 1 K ) . .. I ( 2 , K ) . . . K K ) . . ; 0.0 - 2 6 1 8 . 8 2 2619. 0.0 0.0 -3 .93 29509.2 0.02 - 2 5 0 2 . 9 2 2 50 3. - 7 . 1 6 56406.3 2165. 0.06 - 2 1 6 5 . 4 7 -9.08 1636. 1 6 3 6 . 3 3 0. 12 78310.5 -9.28 962. -962.36 0. 20 93283.0 -7.54 203. 2 0 3 . 2 0 0. 26 99998.4 - 2 6 6 . 7 5 100004.0 2 03. 2 0 3 . 2 0 0. 26 -252.88 962. 9 6 2 . 4 0 2.30 93288.3 - 2 1 7 . 7 2 78315.0 4.15 - 1 6 3 6 . 4 1 1636. - 1 6 4 . 2 3 56409.8 5.65 - 2 1 6 5 . 5 7 2166. - 9 6 . 8 7 29511.2 6.68 - 2 5 0 3 . 0 4 2503. -21.28 7.14 - 2 6 1 8 . 9 6 2619. 21.3 -86.95 6.77 - 2 5 3 2 . 2 3 2532. 25599.3 -32.44 7.11 - 2 6 1 6 . 5 4 2617. 4313.4 -25.02 26. -0.07 9277.2 - 2 5 . 9 4 -868.91 26. 0 . 0 7 9315.8 - 2 5 . 9 4 -868.91 26. -0.07 -25.94 9315.8 -974.06 29. 1 5 . 6 8 9293.4 - 2 4 . 3 8 -1072.37 39. -31.18 9207.0 - 2 2 . 6 6 -1163. 15 5 1. 4 6 . 4 6 9057.1 - 2 0 . 7 7 -1245.75 64. 6 1 . 4 2 8844.9 - 1 8 . 7 4 -1319.61 78. -75.94 8571.9 - 1 6 . 5 8 -1319.61 2 6 0 . 3 4 12. 15 261. 8571.9 -1256.34 -275.34 14.62 276. 7307.6 -1181 .69 -287.84 16.95 288. 5980.5 -1096.34 -297.74 19.13 4605.6 298. -1001.06 -304.93 21.14 3203.7 306. -896.71 -309.36 22.96 1821.8 310. -784.24 -310.99 24.56 792.5 312. -664.65 -309.80 25.95 1512.8 311. -539.04 -305.80 27. 10 2870.6 307. -408.54 -299.03 28.01 4274.2 300. -274.33 -289.54 28 .66 5657.7 291. -137.62 277.44 29.06 6997.7 279. 0.35 -262.81 2 9 . 19 8278.1 264. 138.33 -245.81 29.06 9485.9 248. 275.07 10609.3 -226.58 28.66 228. 409.33 -205.29 28 .01 11637.6 207. 409.33 11637.6 393.97 6.92 394. 436.53 411.83 6.22 9989.3 412. 460.80 426.91 5.47 8274.5 427. 481.95 439.10 4.70 6505.6 439. 499.86 448.32 3.88 4696.7 448. 514.39 454.51 3.05 2867.1 455. 525.45 457.63 2. 19 1087.5 458. 532.96. 457.65 1.31 1065.6 458. 536.86 454.58 0.43 2842.5 455. 537.13 448.44 -0.46 4672.1 448. 533.77 439.27 -1.35 6481.5 439. 526.78 427. 12 -2.22 8251.0 427. 516.23 412.09 -3.08 9966 .6 412. 502 .17 11615.9 394.27 -3.93 394. 484.70 13187.2 373.79 -4.74 374. 463.94 350.78 - 5 . 53 14669.9 351. 463.94 14669.9 -350.43 0.0 350. 463.94 0.0 14669.9 16.62 17. 463.94 14669.9 0.00 16.62 17.  . P ( l f K ) . . . P ( 2 , K ) .. P ( K ) . . . 0.0 0.0 0. 10288.39 0.0 10288. 8561. 8560.88 0.0 5704.00 0.0 5704. 2706.86 0.0 2707. 607.17 0.0 607. . 2500.00 0.0 2500. 607.21 0.0 607. 2707.12 0.0 2707. 5704.57 0.0 5705. 8561.80 0.0 8562. 10289.50 0.0 10290. 0.0 0.0 0. 0.0 0.0 0. 2.95 0.0 3. 0.0 0.0 0. 0.0  0.0  0.27 0.43 0.76 1.27 1 .93 0.0 31.17 34.48 37.29 39.53 4 1 .09 41.94 4 2 .03 41.36 39.96 37.88 35.19 32.00 28.41 24.55 20 .57 0.0 6 0 .75 65.81 70.15 73.66 76 .23 77.81 78.35 77.82 76 .26 73.70 70.20 65.87 60 .82 55.19 49.12 0.0 0.0 0.0  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 .0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0. 0.  0. 1. 1. 2. 0. 31. 34. 37. 40. 41. 42. 42. 41 . 40. 38. 35. 32. 23. 25. 21. 0. 61. 66. 70. 74. 76. 78. 78. 78. 76. 74. 70. 66. 61. 55. 49. 0. 0. 0.  - 214 0. 10288. 18849. 24553. 27260. 27867. 30367 . 30975. 33682. 39386. 47948. 58238. 58238. 120313. 120316. 120316. 120316. 120316. 120316. 120317. 120318. 120320. 120320. 120351. 120386. 120423. 120463. 120504. 120546 . 120588. 120629. 120669. 120707. 120742. 120774. 120802. 120827 . 120847. 120847 . 120908. 120974. 121044. 121118. 121194. 121271. 121350. 121428. 121504. 121578. 121648. 121714. 121774. 121830. 121879. 121879. 121879. 121879.  0.0 0.0 84638.1 11.8 168795.1 26.0 47.9 249764.2 96.9 319210.2 358831.2 492. 1 358872.3 -492.1 305630.6 -96.9 196689.8 -47.9 86266.8 -26.0 19162.8 -11.8 0.0 -0.4 14009.0 -10. 1 343.2 -1.6 3 5 7 . 7 * * >V -J- .-L. -J* 360. 6 3964.8 360.6 3964.8 358.9 -712.7 352.3 -325.0 340.9 -208.6 325 . 1 -152.1 305.3 -118.3 305.3 -33.1 221.9 -26.7 -20.9 148.5 88.1 -15.7 42.6 -10.8 13.8 -6.5 2.6 -11.5 9.5 5.7 34.2 9.7 75.7 14.5 132.6 19.7 202.8 25.3 283.7 31.5 372.4 38.5 4 6 5 .8 46.7 560.3 56.5 560.3 -29.6 412.7 -24.3 -19.4 283.0 174.8 -14.9 91.1 -10.5 33.9 -6.4 4.9 -2.7 4.7 2.7 33.3 6.4 8 9.8 10.5 172.8 14.8 279.8 19.4 408. 1 24.2 554.0 29.5 713.7 35.3 882.9 41.9 41.9 0.0 882. 9 0.0 882.9 -0.0  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  STN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  ZSR. 0,0 0.0016 0.0040 0.0092 0.0294 0.6749 0.6749 0.0307 0.0116 0.0079 0.0073 0.0081 0.0073 0.0079 357.6799 357.6887 357.6887 286.3120 161.9628 92.8928 58.3561 39.8273 3.5428 3.1661 2.8960 2.7057 2.5787 2.5044 2.4773 2.4954 2.5598 2 .6755 2.8517 3.1033 3.4545 3.9435 4.6331 5.6303 1.5567 1.4254 1.3273 1.2555 1.2051 1 .1733 1.1581 1.1588 1.1752 1.2084 1.2602 1.3336 1.4334 1.5666 1.7437 1.9806 1.9806 1.9806 882.8623  ....ZSI . . . . 0.0 11 .7899 26.0481 47.8573 96.9316 492.1165 -492.1443 -96.9329 -47.8577 -26.0483 -11.7901 -0.0002 -10.1093 -1.6485 0.0000 32.5362 32.5362 -144.1833 -175.5665 -151.7822 -124.7655 -102.8337 -32.6989 -26 .3133 -20.5378 -15.1979 -10.1590 -5.3118 -0.5620 4.1776 8.9934 13.9775 19.2348 24.8927 31.1143 38.1203 46.2226 55.8856 -29.4937 -24.2112 -19.3354 -14.7618 -10.4062 -6.1980 -2.0751 2.0195 6.1415 10.3482 14.7014 19.2714 24.1424 29.4185 35.2341 41.7691 41.7691 41.7691 -0.0130  . .ZS 0.0 11.79 26.05 47.86 96.93 492.12 492.14 96.93 47.86 26.05 11.79 0.01 10.11 1.65 . 357.68 359.17 359.17 320.57 238.86 177.95 137.74 110.28 32.89 26.50 20.74 15.44 10.48 5.87 2.54 4.87 9.35 14.23 19.45 25.09 31.31 38.32 46.45 56.17 29.53 24.25 19.38 14.82 10.48 6.31 2.38 2.33 6.25 10.42 14.76 19.32 24.18 29.46 35.28 41.82 41.82 41.82 882.86  PHASE 90.0000 89.9919 89.9911 89.9890 89.9825 89.9214 -89.9214 -89.9818 -89.9860 -89.9827 -89.9647 -1.1866 -89.9586 -89.7247 0.0000 5.1975 5.1975 -26.7293 -47.3080 -58.5327 -64.9332 -68.8287 -83.8163 -83.1389 -81.9738 -79.9051 -75.7574 -64.7571 -12.7821 59.1492 74.1119 79.1636 81.5670 82.8937 83.6646 84.0938 84.2761 84.2470 -86.9786 -86.6305 -86.0728 -85.1388 -83.3941 -79.2806 -60.8345 60.1530 79.1670 83.3394 85.1007 86.0414 86.6021 86.9517 87.1668 87.2.852 87.2852 87.2852 -0.0008  VPHASE 0.0 359.9922 359.9927 359.9932 359.9941 359.9956 180.1528 180.1553 180.1593 180.1668 180.1881 268.9695 180.1946 180.4309 180.1545 185.3519 185.3519 186.0163 186.6886 187.3785 188.0967 188.8556 188.8556 189.8996 191.3961 193.7712 198.2080 209.4865 261.7341 333.9375 349.1768 354.5151 357.2207 358.8730 0.0024 0.8355 1.4857 2.0157 2.0157 2.5046 .3.1924 4.2485 6.1095 10.3354 28.8917 149.9888 169.1131 173.3983 175.2762 176.3395 177.0310 177.5222 177.8936 178. 1877 0.0 0.0 0.0  - 216 TRAP TRITUNE 3 DATE 01-05-72 20:09:50 CENTRE REGION 8 S E C T I O N S IN 2 DEES •RESONATORS* MHZ RESONANT FREQUENCY = 23.1000 OHMS C H A R A C T E R I S T I C IMPEDANCE = 38.30 PFARAD T I P TO T I P C A P A C I T A N C E = 7.00 VOLTAGE,CURRENT PEAKS,POWER L O S S RMS AT S H O R T X = 0 V O L T A G E TO C U R R E N T P H A S E 9 0 D E G R E E S METERS HOT ARM L E N G T H = 3.08622 METERS T I P TO T I P D I S T A N C E = 0.1524 METERS A V E R A G E W I D T H OF S E C T I O N = 1.0000 VOLTS MAX.VOLTAGE ON R E S # 1 = 99998.4 VOLT'S M A X . V O L T A G E ON R E S # 2 = 100004.0 DEGREES VOLTAGE PHASE S H I F T = 179.84 AMPS MAX.CURRENT IN RES#1 = 2618.8 AMPS MAX.CURRENT IN RES#2 = 2619.0 AMPS/CM C U R R E N T D E N S I T Y AT ROOT = 26.188 WATTS POWER L O S S DUE TO B E A M = 0. WATTS POWER L O S S I N R E S O N A T O R S = 115313. WATTS POWER L O S S I N R E S # 1 = 57479. W ATTS POWER L O S S I N R E S # 2 = 57834. •COUPLING LOOP* V O L T A G E INDUCED IN LOOP = LOOP D I M E N S I O N S ARE H I G H T = LENGTH = WIDTH = THICKNESS = LOOP P O S I T I O N = LOOP S E L F INDUCTANCE = CURRENT THROUGH LOOP = POWER L O S S I N L O O P = •TRANSMISSION L I N E * C H A R . I M P E D A N C E OF L I N E = OUTER D I A M E T E R = INNER DIAMETER = MAX.VOLTAGE WITHOUT C P ( 3 ) = M A X . C U R R E N T WITHOUT C P ( 3 ) = MAX.VOLTAGE WITH C P ( 3 ) = MAX.CURRENT WITH C P ( 3 ) = CURRENT THROUGH C P ( 2 ) = C A P A C I T O R A F T E R LOOP C P ( 2 ) = P O S I T I O N OF C P ( 2 ) • = CURRENT THROUGH C P ( 3 ) = CAPACITOR C P ( 3 ) = P O S I T I O N OF CP(3) = LENGTH AFTER C P ( 3 ) = T O T A L L E N G T H OF L I N E = VSWR W I T H O U T C P ( 2 ) = VSWR WITHOUT C P ( 3 ) = VSWR A F T E R C P ( 3 ) = POWER L O S S I N L I N E = POWER L O S S I N N E R C . = POWER L O S S O U T E R C . = •POWER T U B E * TUBE CURRENT = TUBE V O L T A G E = C U R R E N T TO V O L T A G E P H A S E = CAPACITOR C P ( 6 ) AFTER LINE = CURRENT THROUGH C P ( 6 ) = T O T A L POWER L O S S =  9277.2 1.75 17.50 3.00 0.50 3.50 0.224169 24.86 351.6  VOLTS INCHES INCHES INCHES INCHES INCHES MICROHENRY AMPS WATTS  49.78 11.750 5.125 15528. 312. 22802. 458. 186.45 150.000 0.867 600. 355.000 3.8200 2.5500 6.3700 7.55 20.98 44.77 1562. 1088. 474.  WATTS WATTS WATTS  15.94 14669.1 -0.17 164.581 350.4 116878.  AMPS VOLTS DEGREES PFARAD AMPS WATTS  OHMS INCHES INCHES VOLTS AMPS VOLTS AMPS AMPS PFARAD METERS AMPS PFARAD METERS METERS METERS  - 217 •RESONATORS* AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 1 AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 2 QUALITY COMPUTED QUALITY MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA  1 1  = =  2 2  = =  FORESHORTENING CONDUCTIVITY S K I N DEPTH ALFA IN RESONATOR L U M P E D C A P A C I T A N C E AT V O l = LUMPED I N D U C T A N C E AT V O l L U M P E D C A P A C I T A N C E AT V 0 2 = L U M P E D I N D U C T A N C E AT V 0 2 L U M P E D C A P A C I T A N C E AT 1 0 1 = L U M P E D I N D U C T A N C E AT 1 0 1 L U M P E D C A P A C I T A N C E AT 1 0 2 = LUMPED I N D U C T A N C E AT 1 0 2 EQUIVALENT VOLTAGE PEAK 1 = EQUIVALENT VOLTAGE PEAK 2 = I ON O R B I T I N G F R E Q U E N C Y ACCELERATING TIME PERIOD TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN Q M E A S = S H U N T R E S I S T A N C E 1 AT V P E A K = SHUNT R E S I S T A N C E 2 AT V P E A K = SHUNT R E S I S T A N C E AT V L O O P = D E E TO D E E C A P A C I T A N C E AVERAGE ENERGY IN C P ( 1 ) T 0 T = R E A C T A N C E OF C T I P T O T A L D I S T R I B U T E D C A P A C I T Y DEE D I S T R I B U T E D I N D U C T A N C E DEE = RESISTANCE DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA I M A G . C O M P O N E N T OF Z R E S 2 * L 0 0 P AREA/RESONATOR AREA RESONATOR S T E P U P R A T I O DEE S E R I E S R E S I S T A N C E POWER L O S S I N ROOT 1 P O W E R L O S S I N ROOT 2 M U T U A L I N D U C T A N C E AT LOOP C O U P L I N G C O E F F . A T LOOP MUTUAL I N D U C T A N C E D I S T R I B . M U T U A L I N D U C T A N C E AT T U B E C O U P L I N G COEFF.AT TUBE  =  = = =  1.42109 1.42109 2.84 218 1.42125 1.42124 2.84249 7 1 5 5 .14 7155.14 7499.52 23.7160 12.98702 3.24675 145.14143 4.4501 5 .800 0.00137738 0.00003268 568.46 0.0835065 568.46 0.0835062 916.36 0.0518025 916.36 0.0518028 99998. 100004. 4.6200 0.0 98.60 0.0 115313. 86985.44 86461.38 373. 1890 2 8 . 00 0.2800062 -24 6.07 348.129 0.0319167 0.0025035 0.1016 0.1016 0.39370 0.627119 0.4838048 -0.0025873 0.089112 10.77951 0.0016939 1744.44 1744.63 0.0077468 0.05662043 0.0062072 0.0120856 0.00193867  JOULES JOULES JOULES JOULES JOULES JOULES  MHZ METERS METERS 10**6RAD/SEC DEGREES 1 0 * * 7 MHOS CM 1/M PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY VOLTS VOLTS MHZ MICROSEC MICROSEC WATTS OHMS OHMS OHMS PFARAD JOULES OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS METER S**2 1/M OHMS  OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY  - 218 •COUPLING LOOP* 0.0087623 LOOP R E S I S T A N C E 32.53622 LOOP REACTANCE 0.019758 LOOP AREA 3.262 INPUT CURRENT D E N S I T Y 0 . 0 8 1290 LOOP SURFACE 0 . 0 0 0 0346 AVERAGE ENERGY I N LOOP 2.71 L O S S D U E TO I N P U T C U R R E N T = 3 4 8 .93 L O S S D U E TO C A V I T Y C U R R E N T = •-TRANSMISSION L I N E * I M A G . C O M P O N E N T OF Z T L -0.0045431 A L F A IN L I N E 0.00004415 DISTRIBUTED CAPACITY 66 . 9 5 6 9 DISTRIBUTED INDUCTANCE 0.165944 RESISTANCE PER UNIT LENGTH = 0.004396 OUTER RADIUS 0.1492248 INNER RADIUS 0.0650874 AVERAGE ELECTRIC ENERGY 0.0295427 AVERAGE MAGNETIC ENERGY 0.0295444 AVERAGE TOTAL ENERGY 0.0590871 TL Q U A L I T Y FACTOR 5489.29 LUMPED C A P A C I T A N C E AT VTUBE= 0. 1 0 2 0 0 LUMPED I N D U C T A N C E AT V T U B E = 465.38086 S E R I E S R E S I S T A N C E OF LINE = 12.3051 FIRST SECTION AVERAGE MAGNETIC ENERGY 0.0000863 AVERAGE E L E C T R I C ENERGY 0.0011944 AVERAGE TOTAL ENERGY 0.0012807 POWER L O S S O U T E R C O N D U C T O R = 1.3 POWER L O S S I N N E R C O N D U C T O R = 3.1 POWER L O S S T O T A L 4.4 REFLECTION COEFFICIENT 0.766 R E A C T A N C E OF C P ( 2 ) -45 .93 AVERAGE ENERGY IN C P ( 2 ) 0.0027503 SECOND S E C T I O N AVERAGE MAGNETIC ENERGY 0.0099498 AVERAGE E L E C T R I C ENERGY 0.0019950 AVERAGE TOTAL ENERGY 0.0119448 POWER L O S S O U T E R C O N D U C T O R = 160.0 POWER L O S S I N N E R C O N D U C T O R = 3 6 6 .7 POWER L O S S T O T A L 5 2 6 .7 REFLECTION COEFFICIENT 0.909 R E A C T A N C E OF C P ( 3 ) -19.41 IN I T . V O L T A G E M A X . P O S I T I O N = 5.317 THIRD SECTION AVERAGE MAGNETIC ENERGY 0.0194736 AVERAGE ELECTRIC ENERGY 0.0027307 AVERAGE TOTAL ENERGY 0.0222043 POWER L O S S O U T E R C O N D U C T O R = 313.2 POWER L O S S I N N E R C O N D U C T O R = 718.0 POWER L O S S T O T A L 1031.2 REFLECTION COEFFICIENT 0.956 P O S I T I O N OF V O L T A G E M A X I M U M = 8. 1 7 0 AVERAGE ENERGY IN C P ( 3 ) 0.0120186 *POWER T U B E * 920.5391 R E S I S T A N C E TO M A T C H R E A C T A N C E TO M A T C H 41.3573 R E A C T A N C E OF C P ( 3 ) -41.8629 R E S I S T A N C E OF T U B E 920.5320 AVERAGE ENERGY IN C P ( 6 ) 0.0088537  OHMS OHMS METERS**2. AMPS/CM METER**2 JOULES WATTS WATTS OHMS 1/M PFARAD/M MICROHENRY/M OHMS/M METERS METERS JOULES JOULES JOULES PFARAD MICROHENRY OHMS JOULES JOULES JOULES WATTS WATTS WATTS OHMS JOULES JOULES JOULES JOULES WATTS WATTS WATTS OHMS METERS JOULES JOULES JOULES WATTS WATTS WATTS METERS JOULES OHMS OHMS OHMS OHMS JOULES  STN. .METERS., 0.0 I 0.6172 2 0.6172 3 0.6172 4 0.6172 5 0.6172 6 0.0 7 8 0.6172 9 0.6172 10 0.6172 11 0.6172 12 0.6172 13 0.0 14 0.0 15 0.0 16 0.0 17 0.0 18 0.1734 19 0.1734 20 0.1734 21 0.1734 22 0.1734 23 0.0 24 0 . 1969 25 0 .1969 26 0.196 9 27 0.1969 28 0.196 9 29 0.1969 30 0 . 196 9 31 0.1969 32 0 . 196 9 33 0.1969 34 0.1969 35 0.1969 36 0 . 196 9 37 0 .1969 38 0.1969 39 0.0 40 0.1700 41 0.1700 42 0.1700 43 0.1700 44 0.1700 45 0 .1700 46 0.1700 47 0.1700 48 0.1700 49 0.1700 50 0.1700 51 0.1700 52 0.1700 53 0.1700 54 0.1700 55 0.0 56 0.0 57 0.0  . . E (1 , K ) . . , 0.0 29509.17 56406.30 78310 .50 93283.00 99998.44 •100003.75 -93288 .06 -78314.75 -56409.59 -29511.05 -0 .38 -25599.17 -4313.2 9 -9277.21 -92-75 .25 -9275.25 -9242 .36 -9144.47 -8982 .26 -8756.87 -8469.88 -8469.88 -7198 .91 -5862.69 -4473 .33 -3043.41 -1585.91 -114.02 1358.89 2819.50 4254 .54 5651.02 6996 .28 8278.10 9484 .90 10605.72 11630 .39 11630.40 9979 .80 8261.73 6487 .82 4670.04 2820 .67 952 .25 -922.61 -2791.25 -4641 .00 -6459.39 -8234.10 -9953.15 -11604.90 -13178.20 -14662 .42 -14662.42 -14662 .42 -14662.42  219 -  E (2,K ) . . 0. 0 -3 . 93 - 7 . 16 - 9 . 08 - 9 . 28 - 7 . 54 - 2 6 6 . 75 - 2 5 2 . 88 - 2 1 7 . 72 - 1 6 4 . 23 - 9 6 . 87 -21 . 28 - 8 6 . 95 - 3 2 . 44 - 2 5 . 02 - 8 3 3 . 84 - 8 3 3 . 84 - 9 3 4 . 62 -1028. 83 -1115. 82 -1194. 97 -1265. 72 -1265. 72 -1204. 95 -1133. 26 -1051 . 31 - 9 5 9 . 84 - 8 5 9 . 68 -751 . 72 - 6 3 6 . 95 - 5 1 6 . 40 -391 . 17 - 2 6 2 . 38 -131 . 21 1 . 17 133 . 54 264. 73 393. 53 393 . 53 419. 45 442. 56 462. 68 4 7 9 . 68 493. 45 503 . 88 510. 91 514. 48 514. 5 7 511 . 17 504. 32 494. 04 480. 41 4 6 3 . 52 443 . 47 4 4 3 . 47 443. 4 7 443 . 47  .ET(K) . . 0.0 29509.2 56406.3 78310.5 93283.0 99998 .4 100004.0 93288.3 7 8315.0 56409.8 29511.2 21.3 25599.3 4313.4 9277.2 9312.6 9312.6 9289.5 9202.2 9051.3 8838.0 8563.9 8563.9 7299.0 5 971.2 4595 .2 3191.2 1803.9 760.3 1500.8 2366 .4 4272.5 5657.1 6997.5 8278.1 9485 .3 10609.0 11637.0 11637.0 9988 .6 8273.6 6504.3 4694.6 2863.5 1077.3 1054.6 2338.3 4669.4 6479.6 8249.5 9965.4 11614.8 13186.3 14669.1 14669.1 14669.1 14669.1  1,K) . . I ( 2 , K ) . . . 0.0 -2618.82 0.02 -2502.92 0.06 -2165.47 0.12 - 1 6 3 6 . 3 3 0 . 20 -962.36 0 . 26 -203.20 0 . 26 -203.20 2. 30 -962.40 4.15 -1636.41 5.65 -2165.57 6.68 -2503.04 7.14 -2618.96 6.77 -2532.23 7.11 -2616.54 -24.86 -0.07 -24.86 -0.07 -24.86 -0.07 -23.37 -15.68 -21.71 -31.18 -19.90 -46.46 -17.96 -61.42 -15.88 -75.94 11.67 -260.34 14.04 -275.34 16. 28 -287.84 18.37 -297.74 20. 29 -304.93 22.04 -309.37 23. 58 -310.99 24.91 -309.80 26.01 -305.30 26.88 -299.03 27. 51 -289.55 27.88 -277.44 28.01 -262.82 27.88 -245.81 27. 50 -226.58 26.87 -205.29 6. 59 393.97 5.92 411.83 5.21 426.90 4.46 439.09 3.63 448.32 2.88 454.51 2.05 457.63 1.21 457.65 0 . 36 454.58 -0.49 448.44 - 1 . 33 439.26 - 2 . 17 427.12 -3.00 412.09 -3.80 394.27 - 4 . 58 373.79 -5.33 350.78 3 50.45 0.0 0.0 15.94 0 .05 15.94  I (  I  (K) . . , 2619. 2503. 2165. 1636. 962. 203. 203. 962. 1636. 2166. 2503. 2619. 2532. 2617. 25. 25. 25. 28. 38. 51. 64. 78. 261. 276. 288. 298. 306. 310. 312. 311. 307. 300. 291. 279. 264. 247. 228. 207. 394. 412. 427. 439. 448. 455. 458. 458. 455. 448. 439. 427. 412. 394. 374. 351. 350. 16. 16.  - 220 -  ...P(1,K) . . .P(2,K). . P ( K ). . . 0.0 0.0 0. 0. 10288.39 0.0 10288. 10288. 8560.88 0.0 8561. 18849. 5704.00 CO 5704. 24553. • 2706.86 o.d 27 0 7 . 27260. 607.17 0.0 607. 27867. 0.0 0.0 0. 27867. 607.21 0.0 607. 28475. 2707.12 0.0 2707. 31182. 5704.57 0.0 5705. 36886. 8561.80 0.0 8562. 45448. 10289.50 0.0 10290. 55738. 0.0 0.0 0. 55738. 0.0 0. 115313. 0.0 2.71 0.0 3. 115316. 0.0 0.0 0. 115316. 0.0 0. 115316. 0.0 0.25 0.0 0. 115316. 0.41 0.0 0. 115316. 0.75 0.0 1. 115317. 1 .25 0.0 1. 115318. 1.92 0.0 2. 115320. 0.0 0.0 0. 115320. . 31.17 0.0 31 . 115351. 34.47 0.0 34. 115386. 37.28 0.0 37. 115423. 39.51 0.0 40. 115462. 41.08 0.0 41. 115503. 41 .92 0.0 42. 115545. 42.01 0.0 42. 115587. 41 .34 0.0 41. 115629. 39.94 0.0 40. 115668. 37.86 0.0 38. 115706. 35. 17 0.0 35. 115741. 31 .97 0.0 32. 115773. 28.38 0.0 28. 115802. 24.52 0.0 25. 115826. 20.54 0.0 21. 115847. 0.0 0.0 0. 115847. 60.75 0.0 61 . 115907. 65 .81 0.0 66. 115973. 70.15 0.0 70. 116043. 73 .65 0.0 74. 116117. 76.23 0.0 76. 116193. 77.81 0.0 78. 116271. 78.34 0.0 78. 116349. 77.82 0.0 78. 116427. 76.26 0.0 76. 116503. 73 .70 0.0 74. 116577. 70.20 0.0 70. 116647. 65 .87 0.0 66. 116713. 60.82 0.0 61. 116774. 55.19 0.0 55. 116829. 49.12 0.0 49. 13.6878. 0.0 0.0 0. 116878. 0.0 0.0 0. 116878. 0.0 0.0 0. 116878.  0.0 84638.1 168795.1 249764.2 319210.2 358831.2 358872.3 305630.6 196689.8 86266.8 19162.8 0.0 14009.0 343.2 373.2* 376.0 376.0 374.2 367.2 355.2 338.7 318.0 318.0 230.9 154.5 91.5 44. 1 14. 1 2.5 9.7 35.5 78.9 138. 3 211.5 296.0 388 .5 485.9 584.5 584. 5 430.4 295.1 182.3 94.9 35.3 5.0 4.8 34.6 93.6 180.1 291 .7 425.4 577 .6 744.2 920.5 0.0 920.5 920.5  0.0 11.8 26.0 47.9 96.9 492.1 -492.1 -96.9 -47 .9 -26.0 -11.8 -0.4 -10.1 -1.6 4313.2 4313.2 -701.2 -322.2 -207.3 -151.3 -117.7 -33.0 -26.7 -20.9 -15.6 -10.7 -6.4 -10.9 5.6 9.7 14. 5 19.6 25.3 31.5 38.5 46.7 56.5 -29.6 -24.3 -19.4 -14.9 -10.5 -6.4 -2.7 2.6 6.3 10.5 14.8 19.4 24.2 29.5 35.3 . 41.9 41.9 0.0 2.7  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  - 221 -  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  0.0 0.0016 0.0040 0.0092 0.0294 0.6749 0 .6749 0.0307 0.0116 0.0079 0 .0073 0.0081 0.0073 0.0079 373.1890 373.1975 373.1975 291.2400 159.7585 90.2712 56 .3285 38.3177 3.3961 3.0352 2 .7764 2.5942 2.4725 2.4015 2 .3757 2.3932 2 .4552 2.5664 2 .7356 2.9773 3 .3146 3.7843 4 .4468 5 .4050 1.4924 1.3665 1.2725 1.2036 1 .1554 1.1249 1 .1104 1.1110 1 .1268 1.1587 1 .2083 1.2788 1 .3745 1.5023 1.6721 1.8994 1 .8994 1.8994 920.5320  0.0 11 .7899 26 .0481 47.8573 96 .9316 492.1165 -492.1443 -96.9329 -47.8577 -26.0483 - 1 1 . 7 901 -0.0002 -10.1093 -1 .6485 0.0000 32.5362 32 .5362 -155 .4069 -182 .0290 -154.6522 -126.1115 -103 .5196 -32.6862 -26.3003 -20.5246 -15 .1844 -10.1451 -5 .2974 -0.5468 4.1939 9.0112 13.9971 19.2569 24.9181 31.1443 38.1567 46.2684 55 .9456 -29.4963 -24.2133 -19.3371 -14.7632 -10.4073 -6.1989 -2.0759 2.0189 6.1412 10.3480 14.7014 19.2717 24.1430 29.4194 35.2355 41.7711 41 .7711 41 .7711 2.6655  ZS . . . . 0.0 90.0000 11.79 89.9919 26.05 89.9911 47.86 89.9890 96 .93 89.9825 492.12 89.9214 492.14 -89.9214 96.93 -89.9818 47.86 -89.9860 26.05 -89.9827 11.79 -89.9647 0.01 - 1 . 1866 10. 11 -89.9586 1.65 -89.7247 3 7 3 . 19 0.0000 374.61 4.9826 3 74.61 4.9826 330.11 -28.0846 242.19 -48.7280 179.07 -59.7276 138 .12 -65.9317 110.38 -69.6880 32.86 -84.0682 26.47 -83.4168 20.71 -82.2962 15.40 -80.3048 10.44 -76.3031 5.82 -65.6134 2.44 -12.9608 4.83 60.2896 9.34 74.7591 14.23 79.6101 19.45 81.9148 2 5 . 10 83.1864 31.32 83.9250 38.34 84.3360 46.48 84.5101 56.21 84.4816 29.53 -87.1035 24.25 -86.7697 19.38 -86.2350 14.81 -85.3391 10.47 -83.6651 6.30 -79.7146 2.35 -61.8578 2.30 61.1758 6.24 79.6025 10.41 83.6111 14.75 85.3012 19.31 86.2036 2 4 . 18 86.7415 29.46 87.0768 35 .28 87.2830 41.81 87.396 5 41.81 87.3965 41.81 87.3965 920.54 0.1659  . . . V P H A SE 0.0 359.9922 359.9927 359.9932 359.9941 359.9956 180.1528 180.1553 180.1593 180.1668 180.1881 268.9695 180.1946 180.4309 180.1545 185.1371 185.1371 185.7743 186.4193 187.0813 187.7706 188.4993 188.4993 189.5020 190.9404 193.2255 197.5044 208.4609 261.3748 334.8862 349.6211 354.7468 357.3416 358.9255 0.0081 0.8066 1.4298 1.9379 1.9379 2.4067 . 3.0662 4.0792 5.8646 9.9229 27.8853 151.0240 169.5565 173.6732 175.4752 176.4952 177.1584 177.6295 177.9856 178.2676 0.0 0.0 0.0  - 222 -  Appendix B:  CENTRE REGION CYCLOTRON RF SYSTEM PARAMETERS THIRD HARMONIC  - THE RF  - 223 TRAP TRITUNE 3 DATE 02-26-72 13:09:35 CENTRE REGION 8 S E C T I O N S IN 2 DEES •RESONATORS* RESONANT FREQUENCY = 69.3000 M H Z C H A R A C T E R I S T I C IMPEDANCE = 38.30 OHMS T I P TO T I P C A P A C I T A N C E = 7.11 PFARAD V O L T A G E , C U R R E N T P E A K S , P O W E R L O S S RMS AT S H O R T X = 0 V O L T A G E TO C U R R E N T P H A S E 9 0 D E G R E E S HOT A R M L E N G T H = 3.08622 METERS METERS T I P TO T I P D I S T A N C E = 0.1524 METERS A V E R A G E W I D T H OF S E C T I O N = 1.0000 VOLTS M A X . V O L T A G E ON R E S # 1 = 12992.5 VOLTS M A X . V O L T A G E ON R E S # 2 = 13008.0 DEGREES VOLTAGE PHASE S H I F T = 179.91 AMPS MAX.CURRENT IN RES#1 = 348.7 A MPS MAX.CURRENT IN RES#2 = 349.0 A MPS/CM C U R R E N T D E N S I T Y A T ROOT = 3.487 W A TTS POWER L O S S D U E TO B E A M = 0. W A TTS POWER L O S S I N R E S O N A T O R S = 3532. W A TTS POWER L O S S I N R E S # 1 = 1762. W A T TS POWER L O S S I N R E S # 2 = 1771. •COUPLING LOOP* V O L T A G E INDUCED IN LOOP = 1249.1 VOLTS LOOP D I M E N S I O N S ARE HIGHT = 1.75 INCHES INCHES LENGTH = 6, 0 0 INCHES WIDTH = 3. 0 0 0.50 INCHES THICKNESS = INCHES LOOP P O S I T I O N 3.50 MICROHENRY LOOP S E L F INDUCTANCE = 0, 0 8 6 6 7 1 AMPS CURRENT THROUGH LOOP 5.66 WATTS POWER L O S S I N L O O P 5.0 •TRANSMISSION LINE* C H A R . I M P E D A N C E OF L I N E 49.97 OHMS • 5.980 OUTER D I A M E T E R INCHES INNER D IAMETER 2.600 INCHES 2426. VOLTS MAX.VOLTAGE WITHOUT C P ( 3 ) = AMPS MAX.CURRENT WITHOUT C P ( 3 ) 49. VOLTS MAX.VOLTAGE WITH C P ( 3 ) 2426. AMPS MAX.CURRENT WITH C P ( 3 ) 49. AMPS CURRENT THROUGH C P ( 2 ) 34.06 PFARAD C A P A C I T O R A F T E R LOOP C P ( 2 ) = 65 . 0 0 0 METERS P O S I T I O N OF C P ( 2 ) 0. 2 5 4 AMPS CURRENT THROUGH C P ( 3 ) 0. PFARAD CAPACITOR C P ( 3 ) 0.000 M ETERS P O S I T I O N OF C P ( 3 ) 4.1050 M ETERS LENGTH AFTER CP<3) 1.0000 M E TERS T O T A L L E N G T H OF L I N E 5. 1050 VSWR W I T H O U T C P ( 2 ) VSWR W I T H O U T C P ( 3 ) VSWR A F T E R C P ( 3 ) POWER L O S S I N L I N E POWER L O S S I N N E R C . POWER L O S S O U T E R C . •-POWER T U B E * TUBE CURRENT TUBE VOLTAGE C U R R E N T TO V O L T A G E P H A S E CAPACITOR C P ( 6 ) AFTER LINE CURRENT THROUGH C P ( 6 ) . T O T A L POWER L O S S  4.56 16 . 6 6 16.46 47. 33. 14.  =  11.97 598. 3 -0.00 175.044 4 5 .6 3580.  WATTS WATTS WATTS AMPS VOLTS DEGREES PFARAD AMPS WATTS  - 224 •RESONATORS* AVERAGE MAGNETIC ENERGY 1 = AVERAGE E L E C T R I C ENERGY I = AVERAGE TOTAL ENERGY 1 AVERAGE MAGNETIC ENERGY 2 = AVERAGE E L E C T R I C ENERGY 2 = AVERAGE TOTAL ENERGY 2 Q U A L I T Y COMPUTED QUALITY MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA FORESHORTENING CONDUCTIVITY SKIN DEPTH ALFA IN RESONATOR L U M P E D C A P A C I T A N C E AT V O l = LUMPED I N D U C T A N C E AT V O l L U M P E D C A P A C I T A N C E AT V 0 2 = LUMPED I N D U C T A N C E AT V 0 2 L U M P E D C A P A C I T A N C E AT 1 0 1 = LUMPED I N D U C T A N C E AT 101 L U M P E D C A P A C I T A N C E AT 1 0 2 = LUMPED I N D U C T A N C E AT 1 0 2 EQUIVALENT VOLTAGE PEAK 1 = EQUIVALENT VOLTAGE PEAK 2 = ION O R B I T I N G F R E Q U E N C Y ACCELERATING TIME PERIOD TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN O M E A S = SHUNT R E S I S T A N C E 1 AT V P E A K = S H U N T R E S I S T A N C E 2 AT V P E A K = SHUNT R E S I S T A N C E AT V L O O P = D E E TO D E E C A P A C I T A N C E AVERAGE ENERGY IN C P I D T O T = R E A C T A N C E OF C T I P T O T A L D I S T R I B U T E D C A P A C I T Y DEE D I S T R I B U T E D INDUCTANCE DEE = RESISTANCE DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA . BETA IMAG.COMPONENT OF Z R E S 2*L00P AREA/RESONATOR AREA RESONATOR STEPUP RATIO DEE S E R I E S R E S I S T A N C E POWER L O S S I N ROOT 1 POWER L O S S I N ROOT 2 M U T U A L I N D U C T A N C E AT LOOP C O U P L I N G C O E F F . A T LOOP MUTUAL INDUCTANCE D I S T R I B . MUTUAL I N D U C T A N C E AT T U B E C O U P L I N G C O E F F . A T TUBE.  =  =  = = =  0.02515 0.02515 0.05029 0.02521 0.02520 0.05041 12413.29 12413.29 12989.54 7 1.1481 4.32901 1.08225 435 .42432 13.3501 5 .800 0.00079523 0.00005661 595.92 0.0088509 595.90 0.0088511 101.98 0.0517211 101.97 0.0517266 12992. 13008 . 4.6200 0.0 57.02 0.0 3532. 47904.48 47784.64 220.8308 28.44 0.0048072 -80.74 348.129 0.0319167 0.0043362 0.1016 0. 1 0 1 6 0.39370 0.627120 1.4514141 -0.0014938 0.030553 10.41432 0.0003105 53.55 53.67 0.0008499 0.03068500 0.0021282 0.0004020 0.00157798  JOULES JOULES JOULES JOULES JOULES JOULES  MHZ METERS METERS 10**6RAD/SEC DEGREES 1 0 * * 7 MHOS CM 1/M PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY VOLTS VOLTS MHZ MICROSEC MICROSEC WATTS OHMS OHMS OHMS PFARAD JOULES OHMS P FARAD/M MICROHENRY/M OHMS/M METERS METERS METERS MET E R S * * 2 1/M OHMS  OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY  - 225 •COUPLING LOOP* 0.0068657 LOOP R E S I S T A N C E 37.73866 LOOP REACTANCE 0.006774 LOOP AREA 0.742 INPUT CURRENT D E N S I T Y 0.036774 LOOP S U R F A C E 0.0000007 A V E R A G E ENERGY I N LOOP 0.11 L O S S D U E TO I N P U T C U R R E N T = 4.85 L O S S D U E TO C A V I T Y C U R R E N T = •TRANSMISSION LINE* -0.0051652 I M A G . C O M P O N E N T OF Z T L 0.00015001 ALFA IN L I N E 66.7007 DISTRIBUTED CAPACITY 0. 1 6 6 5 8 2 DISTRIBUTED INDUCTANCE 0.014994 R E S I S T A N C E PER UNIT LENGTH = 0.0759460 OUTER RADIUS 0.0330200 INNER RADIUS 0.0002624 AVERAGE ELECTRIC ENERGY 0.0002625 AVERAGE MAGNETIC ENERGY 0.0005249 AVERAGE TOTAL ENERGY 4850.12 TL Q U A L I T Y F A C T O R 0.71925 LUMPED C A P A C I T A N C E AT V T U B E = 7.33317 LUMPED I N D U C T A N C E AT V T U B E = 0.6583 S E R I E S R E S I S T A N C E OF LINE = FIRST SECTION AVERAGE MAGNETIC ENERGY 0.0000005 AVERAGE ELECTRIC ENERGY 0.0000066 AVERAGE TOTAL ENERGY 0.0000071 POWER L O S S O U T E R C O N D U C T O R = 0.0 POWER L O S S I N N E R C O N D U C T O R = 0. 1 POWER L O S S T O T A L 0.1 REFLECTION COEFFICIENT 0.640 R E A C T A N C E OF C P { 2 ) -35.33 AVERAGE ENERGY IN C P ( 2 ) 0.0000235 SECOND S E C T I O N AVERAGE MAGNETIC ENERGY 0.0001930 AVERAGE ELECTRIC ENERGY 0.0001864 AVERAGE TOTAL ENERGY 0.0003794 POWER L O S S O U T E R C O N D U C T O R = 10.5 POWER L O S S I N N E R C O N D U C T O R = 24.2 34.7 POWER L O S S T O T A L 0.887 REFLECTION COEFFICIENT <A> -<*• -lr -L- .JL. t<# <J- «JU -JU T * -|~ * r - ~«" " V R E A C T A N C E OF C P ( 3 ) INIT.VOLTAGE MAX.POSITION = 6.024 THIRD SECTION AVERAGE MAGNETIC ENERGY 0.0000682 AVERAGE ELECTRIC ENERGY 0.0000303 AVERAGE TOTAL ENERGY 0.0000985 POWER L O S S O U T E R C O N D U C T O R = 3.7 POWER L O S S I N N E R C O N D U C T O R = 8.6 POWER L O S S T O T A L 12.3 REFLECTION COEFFICIENT 0.885 P O S I T I O N OF V O L T A G E M A X I M U M = 6.025 AVERAGE ENERGY I N C P ( 3 ) 0.0000000 •POWER T U B E * 50.0056 R E S I S T A N C E TO M A T C H 13.1202 REACTANCE TU MATCH R E A C T A N C E OF C P ( 3 ) -13.1202 R E S I S T A N C E OF T U B E . .= 50.0056 AVERAGE ENERGY I N C P ( 6 ) 0.0000157 - T T> T  1  T-  J-»  OHMS OHMS METERS**2 AMPS/CM MET E R * * 2 JOULES WATTS WATTS OHMS 1/M P FARAD/M MICROHENRY/M OHMS/.M METERS METERS JOULES JOULES JOULES PFARAD MICROHENRY OHMS  .  JOULES JOULES JOULES WATTS WATTS WATTS OHMS JOULES JOULES JOULES JOULES WATTS WATTS WATTS OHMS METERS JOULES JOULES JOULES WATTS WATTS WATTS METERS JOULES OHMS OHMS OHMS O H M S . ... ...... JOULES  STN. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  .METERS.,  0.0 0.6172 0.6172 0.6172 0.6172 0.6172 0.0 0.6172 0.6172 0 .6 1 72 0.6172 0.6172 0.0 0.0 0.0 0.0 0.0 0.0847 0 .0847 0.0847 0.0 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.1540 0.0 0.1429 0.142 9 0.1429 0.1429 0.1429 0.1429 0.1429 0.0 0.0 0.0  .E ( 1 , K ) . . . 0.0 10425.70 13028.68 5855 .85 -5710.79 -12992 .46 13008.04 5720.51 -5859.26 -13042 .67 -10439.76 -3.59 -4590.22 -1723.73 -1249.05 -1248.85 -1248.85 -1239.40 -1211 .25 -1164.84 -1164.84 -670.70 -143.16 391.50 906 .68 1376.72 1778.23 2091 .22 2300.12 2394.51 2369.71 2226.94 1973.32 1621 .46 11 88 .89 697.13 170.67 -364.29 -881.12 -1354.08 -1759.64 -2077.61 -2292.16 -2392.60 -2373.94 -2237.10 -2237.10 -2010.43 -1697.64 -1312.11 -870.38 -391.36 104.43 595.75 595.75 595.75 595.75  - 226 ( K ) . .. E ( 2 , K ) . . . ET I ( 1» K ) . ..1 ( 2 » K ) . . . I ( K ) . 0.0 0.0 0.0 -348.65 349. -0.70 10425 .7 0.01 -217.85 218. -0.30 13028.7 0.02 76. 41 76. 1.03 5 855 .8 0.02 313.34 313. 1.91 5 710.8 -0.02 315.16 315. 1.05 12992.5 -0.06 80.51 81. 2 0 . 19 13008.0 -0.06 80. 51 81. 11.36 5720.5 -0.45 315.47 315. -5.68 5859.3 - 0 . 52 313.73 314. -18.78 13042.7 - 0 . 22 76.59 77. -18.49 10439.8 0. 25 -218.02 218. -4.91 6.1 0 . 54 -349.05 349. -11.34 4590.2 0.46 -327.82 328. -7.39 1723.7 0.52 -346.13 346. -1 .82 1249.1 -5 .66 -0.01 6. -215.27 1267.3 -5 .66 -0.01 6. -215.27 1267.3 -5.66 -0.01 6. -248.29 1264.0 -5 .09 -3.07 6. -277.57 1242.7 - 4 . 44 -6.09 8. -302.68 1203.5 -3.72 -9.01 10. -302.68 1203.5 4.84 -41.98 42. -241.58 712.9 6.07 -46.10 46. -168.47 221. 1 6.99 -47.93 48. -86.97 401.0 7.56 -47.37 48. -1 . 14 906.7 7.76 -44.46 45. 84.75 1379.3 7.57 -39.33 40. 166.44 1786.0 7.01 -32.24 33. 239.87 2104.9 6.09 -23.55 24. 301.37 2319.8 4 . 88 -13.68 15. 347.90 2419.7 3.42 -3.14 5. 377.14 2399.5 1.79 7.56 8. 387.62 2260.4 0.07 17.89 18. 378.83 2009.3 -1.65 27.32 27. 351.21 1659.1 -3.29 35.40 36. 306.12 1227.7 -4.77 41.71 42. 245.80 739.2 -6.00 45.95 46. 173.26 243.2 -6.95 47.90 48. 92.09 375 .7 -7.54 47.46 48. 6.33 381. 1 -7.76 44. 66 45. -79.74 1356.4 -7 . 6 0 39.64 40. -161.87 1767.1 -7.06 32.65 33. -235.95 2091.0 - 6 . 16 24.03 25. -298.31 2311.5 -4.96 14. 21 15. -345.85 2417.5 -3.52 3.69 5. -376.19 2403.6 -1.89 -7.02 7. -387.83 2270.5 -0.18 -17.37 17. -387.83 2270.5 - 0 . 18 -17.38 17. -381.39 2046.3 1.42 -26.22 26. -358.63 1735.1 2.97 -33.94 34. -320.52 1350.7 4.38 -40.20 40. -268.69 910.9 5.61 -44.75 45. -205 .36 442.0 6.59 -47.38 48. -133.24 169.3 7. 30 -47.97 49. -55 .41 598.3 7 .69 -46.51 47. -55.41 5 98.3 0.0 -45.60 46. -55.41 598.3 11.97 0.0 12. -55.41 598.3 11.97 -0.00 12.  . .  - 227 -  P ( 1 , K > . .. P ( 2 , K ) . . P ( K ) . . . .PSUM... • • • Z P R « * « « • • • • Z P I * » * . .STN 0.0 0.0 0. 0. 0.0 0.0 1 251.26 0.0 251 . 251. 432604.9 47.9 2 35.26 0.0 287. 592442.4 35. -170.5 3 129.94 0.0 416. 130. 82339.5 -18.7 4 304.43 0.0 304. 18.1 721. 45240.0 5 133.27 0.0 133. 854. 197624.6 161.4 6 0.0 0.0 854. 198099.2 0. -161.6 7 133.50 0.0 134. 988. -18.1 33133.0 8 305.13 0.0 305. 26555.6 1293. 9 18.7 130.31 0.0 1423. 119535.8 130. 10 170.3 35.31 0.0 35. 1458. 74731.6 -47.9 11 251.77 0.0 252. 1710. -0.0 0.0 12 0.0 0.0 0. 1710. 13270.1 -14.0 13 0.0 0.0 0. 3532. -5.0 1786.2 14 0.11 0.0 3533. 220.8 682920.5 0. 15 0.0 0.0 3533. 0. 1327.8 227.3 16 0.0 0.0 0. 3533. 1327.8 227.3 17 0.0 0.02 0. 3533. 226.2 -6 2 8.6 18 0.03 0.0 218.6 0. 3533. 19 -251.5 0.0 0.05 3533. 0. 205.0 -154.6 20 0.0 0.0 3533. -28.8 0. 205.0 21 2.30 0.0 2. 3535. 71.9 -15.7 22 2.62 0.0 3. 3538. 6.9 -6.1 23 2.71 0.0 3. 22.7 3540. 9.0 24 2.52 0.0 3. 3543. 116.0 20.4 25 2.11 0.0 2. 3545. 268.4 26 34.7 1 .56 0.0 2. 3546. 449.7 54. 5 27 0.96 0.0 3547. 1. 624.5 87.4 28 0.45 0.0 3548. 0. 758.4 29 163.3 0.11 0.0 3548. 825.1 0. 672.4 30 0.02 0.0 3548. 0. 811.5 -333.9 31 0.20 0.0 3548. 0. 720.1 -128.4 32 0.0 0.60 3549. 1. 568.9 -74.0 33 0.0 1.16 387.7 1. 3550. 34 -47.0 1 .75 0.0 2. 3552. 212.2 -29.5 35 2.27 0.0 3554. 2. 76.9 -16.3 36 2.61 0.0 3. 3557. 8.3 -6.3 37 2.71 0.0 3. 3559. 19.8 8.5 38 2.54 0.0 3. 19.8 39 3562. 109.0 2.14 0.0 2. 3564. 258.1 33.9 40 1.59 0.0 2. 3566. 437.9 41 53.3 0.0 0.99 3567. 1. 613.0 85.1 42 0.0 0.47 3567. 0. 156.9 749.0 43 0.0 0.13 3567. 0. 581 .9 44 819.2 0.02 0.0 3567. 809. 8 0. -362.3 45 0.19 0.0 3567. 722.6 0. -132.9 46 0.0 0.0 0. 3567. 722.6 -132.9 47 0.0 0.52 3568. 1. 586 .8 -78.6 48 0.0 0.99 3569. 1. 421.8 49 -51.3 1.50 0.0 2. 3570. 255.5 -33.7 50 0.0 1 .98 2. 116.1 3572. -20. 5 51 0.0 2.33 2. 3575. 27.3 -9.8 52 2.50 0.0 3577. 3. 7.1 4.0 53 2.47 0.0 2. 3580. 54 50.0 13.1 0.0 0.0 0. 3580. 0.0 13.1 55 0.0 0.0 0. 3580. 50.0 0.0 56 0.0 0.0 0. 3580. 50.0 0.0 57  - 228 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 .57  .  0.0 0 .0053 0 .0491 0.0042 0.0073 0.1318 0.1318 0.0099 0.0131 0.2426 0.0307 0.0140 0.0148 0.0139 2 2 0 .8434 220.8502 2 2 0 .8502 200.2400 124.5143 74.3112 3.9563 3.2695 3.0154 3.0765 3.4789 4.4199 6.5155 11.9901 33 . 6 1 5 9 329.2678 117.4910 22.1775 9.4727 5.6177 4.0303 3.3103 3.0367 3.0823 3.4662 4.3748 6.3914 11 . 5 9 1 5 31 .4775 274.7024 135 . 0 5 1 2 23.6301 23.6297 10.3497 6.1497 4.3656 3.5127 3.1247 3.0382 3.2207 3.2207 3.2207 50.0056  0 .0 4 7 . 8 5 74 -170 .5079 -18.6886 18.1204 161.3861 -161.5797 -18.1330 18.6758 170.2937 -47.8838 -0.0103 -14.0021 -4.9800 0 .0714 3 7.8101 37.8101 -72.0439 -108.2.221 -98.5571 -28.2043 -14.9780 -3 .4263 7.7733 19.7875 34.1556 53.7388 85.7003 156.0949 404.0674 - 2 8 5 .5422 -124.4077 -72 .7953 -46.3297 - 2 8 .9641 -15.6053 -4.0037 7. 1858 19.1256 33.3189 52 .5161 83.4916 150 .2835 386.7505 -301 .8708 -128.5143 -128.5134 -77.2430 - 5 0 .5585 -33.1115 -19.8905 -8.6956 1 .7147 12.2752 12.2752 12.2752 0 . 0 0 0 1 ..  zs....  0.0 4 7 . 86 170.51 18.69 18.12 161.39 161.58 1 8 . 13 18 .68 170.29 47.88 0.02 14.00 4. 98 220.84 224.06 224.06 212.81 164.97 123.43 28.48 15.33 4.56 8.36 20.09 34.44 54.13 86.53 159.67 5 2 1 . 24 308 .77 126.37 73.41 . 46.67 29.24 15.95 5 .03 7.82 19 .44 33.60 52.90 8 4 . 29 153 .54 474.38 330.70 130.67 130.67 77.93 50.93 33.40 20.20 9.24 3 .49 12.69 12.69 12.69 50.01  90.0000 89.9936 -89.9835 -89.9870 89.9770 89,9531 -89.9532 -89.9686 89.9597 89.9183 -89.9633 -36.3040 -89.9395 -89.8402 0.0185 9.7150 9.7150 -19.7881 -40.9956 -52.9840 -82.0150 -77.6862 -48.6500 68.4074 80.0285 82.6266 83.0869 82.0355 77.8466 50.8240 -67.6344 -79.8923 -82.5858 -83.0862 -82.0783 -78.0237 -52.8203 66.7836 79.7275 82.5198 83.0609 82.0959 78.1701 54.6144 -65 . 8 9 7 1 -79.5813 -79.5814 -82.3684 -83.0649 -82.4891 -79.9847 -70.2346 29.4390 75.2984 75 . 2 9 8 4 75.2984 0.0001 . .  VPHAS F 0.0 359.9961 359.9985 0.0101 179.9803 179.9954 0.0889 0.1137 180.0556 180.0825 180.1015 233.7845 180.1416 180.2456 180.0836 189.7800 189.7800 191.3282 192.9072 194.5657 194.5657 199.8088 229.6432 347.4749 359.9277 3.5228 5.3474 6.5434 7.4647 8.2667 9.0427 9.8739 10.8673 12.2215 14.4391 19.4222 45.4315 165.8132 179.5881 183.3704 185.2558 186.4793 187.4151 188.2251 189.0046 189.8352 189.8353 190.7417 191.9285 193.7273 197.1559 207.6880 308.0889 354.6865  0.0 0.0  0.0  - 229 -  Appendix C:  MAIN CYCLOTRON RF SYSTEM PARAMETERS - THE RF FUNDAMENTAL  - 230 TRAP TRITUNE 3 DATE 01-05-72 19:38:53 MAIN CYCLOTRON 80 SECTIONS IN 2 DEES •RESONATORS* RESONANT FREQUENCY = 23.1000 MHZ OHMS CHARACTERISTIC IMPEDANCE = 46.00 PFARAD TIP TO TIP CAPACITANCE = 6.50 VOLTAGE,CURRENT PEAKS,POWER LOSS RMS AT SHORT X=0 VOLTAGE TO CURRENT PHASE 90 DEGREES HOT ARM LENGTH = 3.06780 METERS T I P TO TIP DISTANCE = 0.1524 METERS AVERAGE WIDTH OF SECTION = 0.8320 METERS MAX.VOLTAGE ON RES#1 = 99998.8 VOLTS MAX. VOLTAGE ON RES#2 = 100005.7 VOLTS VOLTAGE PHASE SHIFT = 179.86 DEGREES MAX.CURRENT IN RES#1 = 2182.1 AMPS MAX.CURRENT IN RES#2 = 2182.2 AMPS CURRENT DENSITY AT ROOT = 26.227 AMPS/CM POWER LOSS DUE TO BEAM = 300000. WATTS POWER LOSS IN RESONATORS = 1259600. WATTS POWER LOSS IN RES#1 = 479595. WATTS POWER LOSS INRES#2 -= 4 8 0 0 0 5 . WATTS -•COUPLING LOOP* VOLTAGE INDUCED IN LOOP = 9291.0 VOLTS LOOP DIMENSIONS ARE HIGHT = 1.75 INCHES LENGTH = 17.50 INCHES WIDTH = 3.00 INCHES THICKNESS = 0.50 INCHES LOOP POSITION = 3.50 INCHES LOOP SELF INDUCTANCE = 0.224169 MICROHENRY CURRENT THROUGH LOOP = 271.14 AMPS POWER LOSS IN LOOP = 672.1 WATTS •TRANSMISSION L I N E * CHAR. IMPEDANCE OF LINE = 54.17 OHMS OUTER DIAMETER = 11.100 INCHES INNER DIAMETER = 4.500 INCHES MAX.VOLTAGE WITHOUT C P ( 3 ) = 17399. VOLTS MAX.CURRENT WITHOUT C P ( 3 ) = 321. AMPS MAX.VOLTAGE WITH C P ( 3 ) = 24268. VOLTS MAX.CURRENT WITH C P ( 3 ) = 448. AMPS CURRENT THROUGH C P ( 2 ) = 270.21 AMPS CAPACITOR AFTER LOOP CP{2) = 120.000 PFARAD POSITION OF C P ( 2 ) = 0.610 METERS AMPS CURRENT THROUGH C P ( 3 ) = 303. PFARAD CAPACITOR C P ( 3 ) = 120.000 METERS POSITION OF CP(3) = 31.9200 METERS LENGTH AFTER C P ( 3 ) = 2.4800 METERS TOTAL LENGTH OF LINE = 34.4000 VSWR WITHOUT C P ( 2 ) = 2.36 VSWR WITHOUT C P ( 3 ) = 2.21 VSWR AFTER C P ( 3 ) = 4.29 POWER LOSS IN LINE = 5964. WATTS POWER LOSS INNER C. = 4244. WATTS POWER LOSS OUTER C. = 1720. WATTS •POWER TUBE* TUBE CURRENT = 223.69 AMPS TUBE VOLTAGE = 11318.6 VOLTS CURRENT TO VOLTAGE PHASE = 0.00 DEGREES CAPACITOR C P ( 6 ) AFTER LINE = 208.994 PFARAD CURRENT THROUGH C P ( 6 ) = 343.3 AMPS TOTAL POWER LOSS = 1265886. WATTS  •RESONATORS* AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 1 AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY 2 Q U A L I T Y COMPUTED QUALITY MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA  1 1  = =  2 2  =  FORESHORTENING CONDUCTIVITY SKIN DEPTH A L F A IN R E S O N A T O R L U M P E D C A P A C I T A N C E AT V O l = L U M P E D I N D U C T A N C E AT V O l L U M P E D C A P A C I T A N C E AT V 0 2 = L U M P E D I N D U C T A N C E AT V 0 2 L U M P E D C A P A C I T A N C E AT 1 0 1 = L U M P E D I N D U C T A N C E AT 1 0 1 LUMPED C A P A C I T A N C E AT 102 = L U M P E D I N D U C T A N C E AT 1 0 2 EQUIVALENT VOLTAGE PEAK 1 = EQUIVALENT VOLTAGE PEAK 2 = ION O R B I T I N G F R E Q U E N C Y ACCELERATING TIME PERIOD TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON R E S O N A T O R POWER WHEN Q M E A S = S H U N T R E S I S T A N C E 1 AT V P E A K = SHUNT R E S I S T A N C E 2 AT V P E A K = SHUNT R E S I S T A N C E AT VLOOP= D E E TO D E E C A P A C I T A N C E AVERAGE ENERGY IN C P ( 1 ) T 0 T = R E A C T A N C E OF C T I P T O T A L D I S T R I B U T E D C A P A C I T Y DEE D I S T R I B U T E D INDUCTANCE DEE = RESISTANCE DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA . I M A G . C O M P O N E N T OF Z R E S 2 * L 0 0 P AREA/RESONATOR AREA RESONATOR STEPUP RATIO DEE S E R I E S R E S I S T A N C E POWER L O S S I N ROOT 1 POWER L O S S I N ROOT 2 M U T U A L I N D U C T A N C E AT L O O P C O U P L I N G C O E F F . A T LOOP MUTUAL INDUCTANCE D I S T R I B . M U T U A L I N D U C T A N C E AT T U B E C O U P L I N G C O E F F . A T TUBE  =  = = =  11.84865 11.84862 2 3.6 97 27 11.85037 11.85016 23.70052 7169.01 7169.01 7515.96 23.8549 12.98702 3.24675 145.14143 4.9606 5.800 0.00137738 0.00003271 4739.59 0.0100156 4739.62 0.0100155 7630.28 0.0062212 7630.23 0.0062213 114574. 114571. 4.6200 270.56 98 . 7 9 6250.0 1259600. 10425.20 104 1 7 . 7 4 34.2658 260.00 2.6001081 -26 .50 2898.551 0.0038333 0.0003009 0.1016 0. 1 0 1 6 0.39370 0.623377 0.4838048 -0.0031098 0.089647 10.76373 0.0002028 14556.38 14558 .39 0.0009305 0.01963754 0.0007461 0.00G9846 0.00324607  JOULES JOULES JOULES JOULES JOULES JOULES  MHZ METERS METERS 10**6RAD/SEC DEGREES 1 0 * * 7 MHOS CM 1/M PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY VOLTS VOLTS MHZ MICROSEC MICROSEC WATTS OHMS OHMS OHMS PFARAD JOULES OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS METER S**2  1/M OHMS  OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY  - 232 •COUPLING LOOP* LOOP R E S I S T A N C E 0.0087623 LOOP REACTANCE 3 2.5 36 22 LOOP A R E A 0.019758 INPUT CURRENT DENSITY 35.583 LOOP S U R F A C E 0.081290 A V E R A G E ENERGY I N LOOP 0.0041202 L O S S D U E TO I N P U T C U R R E N T = 322.10 L O S S D U E TO C A V I T Y C U R R E N T = 34 9 . 9 6 •TRANSMISSION LINE* -0.0050632 I M A G . C O M P O N E N T OF Z T L 0.00004522 ALFA IN LINE 61.5347 DISTRIBUTED CAPACITY 0.180567 DISTRIBUTED INDUCTANCE 0.004899 R E S I S T A N C E PER UNIT LENGTH = 0.1409701 OUTER RADIUS 0.0571500 INNER RADIUS 0.1149159 AVERAGE E L E C T R I C ENERGY 0.1149188 AVERAGE MAGNETIC ENERGY 0.2298347 AVERAGE TOTAL ENERGY 5593.32 TL Q U A L I T Y FACTOR 5.16729 LUMPED C A P A C I T A N C E AT V T U B E = 9.18659 L U M P E D I N D U C T A N C E AT V T U B E = 0.2384 S E R I E S R E S I S T A N C E OF LINE = FIRST SECTION AVERAGE MAGNETIC ENERGY 0.0016620 AVERAGE ELECTRIC ENERGY 0.0019031 AVERAGE TOTAL ENERGY 0.0035651 POWER L O S S O U T E R C O N D U C T O R = 25.7 POWER L O S S I N N E R C O N D U C T O R = 63.3 POWER L O S S T O T A L 89.0 REFLECTION COEFFICIENT 0.405 R E A C T A N C E OF C P ( 2 ) . -57.42 AVERAGE ENERGY IN C P ( 2 ) 0.0072206 SECOND S E C T I O N AVERAGE MAGNETIC ENERGY 0.0895221 AVERAGE E L E C T R I C ENERGY 0.0859562 AVERAGE TOTAL ENERGY 0.1754782 POWER L O S S O U T E R C O N D U C T O R = 1388.9 POWER L O S S I N N E R C O N D U C T O R = 3426.1 POWER L O S S T O T A L 4815.0 REFLECTION COEFFICIENT 0.378 R E A C T A N C E OF C P ( 3 ) -57.42 INIT.VOLTAGE MAX.POSITION = 31.980 THIRD SECTION AVERAGE MAGNETIC ENERGY 0.0196147 AVERAGE ELECTRIC ENERGY 0.0040667 AVERAGE TOTAL ENERGY 0.0236814 POWER L O S S O U T E R C O N D U C T O R = 305.8 POWER L O S S I N N E R C O N D U C T O R = 754.2 POWER L O S S T O T A L 1060.0 REFLECTION COEFFICIENT 0.622 P O S I T I O N OF V O L T A G E M A X I M U M = 36.760 AVERAGE ENERGY IN C P ( 3 ) 0.0090759 *POWER T U B E * 50.5991 R E S I S T A N C E TO M A T C H , = R E A C T A N C E TO M A T C H 32.9667 R E A C T A N C E OF C P ( 3 ) -32.9667 R E S I S T A N C E OF T U B E 50.5991 AVERAGE ENERGY IN C P ( 6 ) 0.0066935  OHMS OHMS MET ER S * * 2 AMPS/CM MET E R * * 2 JOULES WATTS WATTS OHMS 1/M PFARAD/M MICROHENRY/M OHMS/M METERS METERS JOULES JOULES JOULES PFARAD MICROHENRY OHMS JOULES JOULES JOULES WATTS WATTS WATTS OHMS JOULES JOULES JOULES JOULES WATTS WATTS WATTS OHMS METERS JOULES JOULES JOULES WATTS WATTS WATTS METERS JOULES OHMS OHMS OHMS OHMS JOULES  STN. 1 2 3 4  0.0 0.6136 0.6136 0.6136  5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  0.6136 0.6136 0.0 0.6136 0.6136 0.6136 0.6136 0.6136 0.0 0.0 0.0 0.0 0.0 0.2033 0.2033 0.2033 0.0 0 .3684 0.3684 0.3684 0.3684 0.3684 0.3684 0 .3684 0.3684 0.3684 0.3684 0.3684 0.3684 0 .3684 0.3684 0 .3684 0.3684 0.3684 0.3684 0.3684 0.3684 0.3684 0.3684 0 .3684 0.3684 0 .3684 0.3684 0.3684 0.3684 0 .3684 0.3684 0.36.84 0.3684 0.3684 0.3684 0.3684 0.3684 0.3684  0.0 29359.91 5 6 1 5 1 .70 7 8031.81 9 3 0 8 6 .44 99998.75 - 1 0 0 0 0 5 .50 - 9 3 0 9 2 .81 - 7 8 0 3 7 .25 -56155.87 - 2 9 3 6 2 .35 -0.55 -25618.55 -4316.61 -9290.97 -9272.03 - 9 2 7 2 .03 -9223.97 - 9 0 8 6 .71 - 8 8 6 1 .56 - 8 8 6 1 .56 -6754.96 -4434.34 -1973.19 5 50 .53 3056.87 5 4 6 6 .41 7702.85 96 95 .32 1 1 3 8 0 .70 1 2 7 0 5 .58 13628.00 14113.71 14162.18 1 3 7 5 6 .99 12 9 1 6 . 0 0 11665 .81 10046.04 8108.00 5913.04 3530.71 103 6.50 - 1 4 9 0 .61 -3970.57 -6324.80 . -8478.70 - 1 0 3 6 4 .07 -11921.15 - 1 3 1 0 0 .63 -13 865.11 -14190.3.8 -14066.11 -13496.25 -12493.82 -11105.44 -9360.20 -7318.39 -5044.71  - 233 ET ( K ) . I.<( 1, K ) . I. ( 2 , K ) . . . 0.0 -2182.06 0.0 0.0 -3.91 2086.63 0.01 29359.9 - 7 . 14 1808.67 0.05 56151.7 -9.08 1 372.52 0, 10 78031.8 816.30 -9.31 0. 16 9 3 0 8 6 .4 -7.63 188.69 0. 22 99998.8 - 2 3 8 . 8 4 L00005 .7 1 8 8 .69 0.22 8 16.35 -226.38 1.73 93093.1 1 3 7 2.60 -195.22 3. 10 78037.4 1 8 0 8 .79 -147.89 4. 21 56156.0 2 0 8 6 .76 -88.29 4.98 29362.5 2 1 8 2 . 20 -21.31 5. 3 4 21.3 2 1 0 9 . 94 -79.85 5.05 25618.7 2 1 8 0 . 1 9 -31 .27 5.31 4316.7 0 . 6 6 -22.45 9291.0 -271.14 - 0 . 66 -8844.43 12813.8 - 2 7 1 . 1 4 -0.66 -8844.43 12813.8 -271.14 1 7.46 - 10244.24 13785.0 -253.80 3 4.10 -11545.02 14692.0 -234.00 5 0 .41 - 1 2 7 3 4 . 17 15514.1 -211.93 2 0 4 .75 9.86 -12734.17 15514. 1 2 3 0 . 51 5 1.37 -12438.21 14154.1 2 4 8 . 97 91.27 -11748.32 12557.3 2 5 9 . 53 128.27 10867.0 - 10686.32 2 6 1 . 88 161.21 -9285.84 9302.1 2 5 5 . 9 3 189.04 8183.6 - 7 5 9 1 .23 2 4 1 . 8 7 210.89 7 8 6 6 .0 -5656.17 2 2 0 . 1 5 226.06 -3541.90 8478.1 -191.46 234.07 97 8 4 . 1 -1315.39 156.70 234.67 11420.5 952.83 116.97 227.8 4 13100.1 3190.95 -73.54 213.79 14632.5 5328.06 -27.78 192.96 15892.6 7296.49 1 8 . 87 166.03 16798.2 9 0 3 3 .90 6 4.91 133.84 17297.2 10485.22 1 0 8.90 97.40 17363.4 11604.52 1 4 9.45 57.89 16993.2 12356.30 1 8 5 .26 1 6 . 53 16206.1 12716.78 2 1 5 . 20 -25.34 15046.0 12674.50 2 3 8 . 32 -66.42 13585.1 12230.81 2 5 3 . 9 0 11934.0 -105.39 11399.74 2 6 1 . 4 3 10207.61 10260. 1 -141.03 260.68 8 8 1 9 . 0 - 1 7 2 . 20 8692.16 2 51,67 7 96 2.0 - 1 9 7 . 9 2 6901.36 2 34.69 4891.94 7995.9 -217.37 2 1 0.28 2727.54 8906.6 -229.94 1 7 9.20 10375.0 -235.22 476.68 1 4 2 .45 -1739.32 12054.7 -233.06 1 0 1 .18 -3998.73 1 3 6 9 7 . 3 -22 3.51 5 6 . 70 15140.2 -206.89 - 6 0 8 1 .56 1 0 . 4 3 16276.2 -183.71 -7971.82 3 6 . 1 7 17035.3 -154.71 -9609.67 8 1 . 6 3 17375.3 -120.81 -10943.21 -124.50 -83.09 17278.6 -11930.20 -163.43 -42.73 16750.1 -12539.38 197.19 -1.02 15818.1 -12751.43 224.69 40.73 14536.3 -12559.64 2 45.09 8 1 . 18 12989.7 -11970.06  I  ( K) .. 2182. 2087. 1809. 1373. 816. 189. 189. 816. 1373. 1 809. 2087. 2182. 2110. 2180. 271. 271. 271. 254. 236. 218. 205. 236. 265. 289. 308. 318. 321. 316. 302. 282. 256. 226. 195. 167. 149. 146. 160. 186. 217. 247, 275. 297. 312. 320. 320. 312. 296. 273. 245. 215. 184. 159. 146. 150. 169. 197. 228. 258.  , p ( l , K ) . . .P(2,K )• • P ( K } . 0.0 0.0 0. 8536.86 0.0 8537. 7119.47 0.0 7119. 0.0 4769.78 4770. 2291.92 0.0 2292. 0.0 534. 533.90 15000.00 0.0 15000. 533.94 0.0 534. 2292.20 0.0 2292. 0.0 4770.39 4770. 7120.43 0.0 7120. 8538.01 0.0 8538. 0.0 0.0 0. 0.0 0.0 0. 322.10 0.0 322. 0.0 0.0 0. 0.0 0.0 0. 34.43 0.0 34. 30.03 0.0 30. 25.72 0.0 26. 0.0 0.0 0. 43 .99 0.0 44. 0.0 56.89 57. 69.67 0.0 70. 80.73 0.0 81. 88 .68 0.0 89. 92.52 0.0 93. 91 .76 0.0 92. 86.50 0.0 86. 0.0 77.40 77. 65.61 0.0 66. 52.61 0.0 53. 40.03 0.0 40. 29.45 0.0 29. 22.20 0.0 22. 19.20 0.0 19. 20.83 0.0 21. 26.87 0.0 27. 36.57 0.0 37. 48.71 0.0 49. 61.76 0.0 62. 0.0 74.09 74. 84.14 0.0 84. 90.65 0.0 91. 92.80 0.0 93. 90.33 0.0 90. 83.54 0.0 84. 73.29 0.0 73. 60.86 0.0 61. 47.82 0.0 48. 0.0 35.81 36. 26 .33 0.0 26. 0.0 20.59 21 . 19.29 0.0 19. 22.61 0.0 23. 30.13 0.0 30. 40.90 0.0 41 . 53.57 0.0 54.  . 0. 8537. 15656. 20426. 22718. 23252. 38252. 38786. 41078. 45848. 52969. 61507. 61507. 1259600. 1259922. 1259922. 1259922. 1259956. 1259986. 1260011. 1260011. 1260054. 1260110. 1260179. 1260259. 1260347. 1260439. 1260530. 1260616. 1260693. 1260758. 1260810. 1260850. 1260879. 1260901. 1260920. 1260940. 1260966. 1261002. 1261050. 1261111. 1261185. 1261269. 1261359. 1261451. 1261541. 1261624. 1261697. 1261757. 1261804. 1261839. 1261865. 1261885. 1261904. 1261926. 1261956. 1261996. 1262 0 4 9 .  0.0 0.0 14.1 100974.3 201388.9 31.0 298096.8 56.9 114.0 381418.3 430059.4 530.0 -530.0 430119.2 364345.4 -114.0 -56.9 233519.4 102226.0 -31.0 22706.9 -14.1 0.0 -0.3 16815.1 -12.1 411.9 -2.0 *X* ^1*. «Jt*< ^JU 3 4 . 3 * * -T* " ¥ " " V * - 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9 4 .86 2424.53 4 8 6 7 .16 7155.66 9217.52 10987.38 12409.31 13438.10 14041 .24 14199.56 1 3 9 0 8 .07 13175.96 1 2 0 2 6 .46 10495.95 8 6 3 2 .86 6496.30 4 1 5 3 .91 1679.79 -847.52 -3348.14 -5742.63 -7955.27 - 9 9 1 6 .04 -11562.62 -12842 .98 -13716.56 -14155 .57 -14146.21 -13688 .66 -12797.51 -11500 .86 -9839.91 - 7 8 6 7 .21 -5645.15 -3244.30 -740.64 1786 .66 4257.32 6 5 9 3 .15 8720.29 10571 .11 12087.18 1 3 2 2 0 .30 13934.67 14207.61 14030.49 1 3 4 0 8 .86 13408.82 1 0 6 6 4 .71 7 6 8 1 .13 4 5 2 5 .05 1267.34 -2018.85 -5259.74 -8382.54 -11317.15 -11317.15 -11317.15 -11317.15  - 235 -11001 .37 11307.0 - 9 6 8 4 .20 96 84 .7 - 8 0 6 0 .30 8417.0 - 6 1 8 1 .07 7867.3 - 4 1 0 6 .02 8 25 0.0 -1900 .89 94 11.5 3 6 4 .44 10993.4 2 6 1 8 .42 12682.5 4 7 8 9 .45 14266.1 6 8 0 8 .93 156 05 . 1 8 6 1 2 .76 16607.4 10143 .86 17214.3 1 1 3 5 3 .78 17392.9 1 2 2 0 4 .11 17134.0 1 2 6 6 7 .95 16451.2 1 2 7 3 0 .60 15381.6 1 2 3 9 0 .05 13989.8 1 1 6 5 7 .11 12375.1 1 0 5 5 4 .91 10687.7 9118 .43 9157.7 7 3 9 3 .04 8 115.9 5 4 3 3 .55 7905.8 3301 .92 8613.3 1065 .55 9973.1 - 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236 1262115. 1262193. 1262280. 1262372. 1262464. 1262552. 1262632. 1262700. 1262756. 1262799. 1262830. 1262853. 1262872. 1262892. 1262916. 1262949. 1262994. 1263052. 1263123. 1263204. 1263293. 1263385. 1263476. 1263561. 1263637. 1263701. 1263752. 1263790. 1263818. 1263839. 1263858. 1263879. 1263906. 1263943. 1 263993. 1264056. 1264131. 1264216. 1264307. 1264399. 1264488. 1264570. 1264642. 1264701. 1264747. 1264781. 1264806. 1264826. 1264826. 1264919. 1265032. 1265162. 1265305. 1265455. 1265606. 1265752. 1265886. 1265886. 1265886. 1265886.  50.6 37.2 28. 1 24.5 27.0 35.1 47.9 63.7 80.6 96.4 109. 2 117.3 119.8 116.2 107. 1 93.7 77.5 60.6 45.2 33.2 26.1 24.7 29.4 39.4 53.5 69.9 86.7 101.6 112.8 113.9 119.1 113.4 102. 5 87.8 "7 1 . 2 54.6 40.3 29.9 24.9 25.8 32.5 44.2 59.4 76.2 92.5 106.2 115.6 119.6 119.6 87.2 58.0 34.6 19.0 12.7 16.2 29.2 50.6 0.0 50.6 50.6  -64.5 -62.2 -83.9 -835.1 96.5 63.4 63.2 73.6 93.1 127.4 197.5 419.7 -4120.9 -348.0 -179.5 -119.3 -88.6 -71.0 -62.3 -65.5 -116.5 284.5 75.9 61.7 66.1 79.6 103.6 147.1 246.5 712.5 -822.7 -258.8 -151.6 -105.8 -81.0 -66.8 -61.6 -73.5 -227.2 128.0 66.7 62.1 70.2 87. 1 116.6 173.5 326.5 2326.4 -58.9 -45.3 -35.3 -28.4 -27.8 -146.2 31.4 27.2 33.0 33.0 0.0 -0.0  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  0.0 0 .0020 0.0048 0 .0108 0.0341 0.6531 0.6531 0.0357 0.0138 0 .00 94 0.0087 0 .0098 0.0088 0 .00 95 34.2659 34.2746 34.2746 38 . 9 3 6 9 45.0662 53 . 1013 59.9696 45 .1838 35.8429 30 .0724 26.6529 24.8990 24.4800 2 5 . 31 94 27.5709 31 .6663 38.4429 49.3352 66.3495 90.3155 113.9754 118 . 1 3 2 3 98.1854 72.9036 53.7151 41.2053 33.3 75 6 28.5872 25.8439 24.6093 24.6546 25.9879 28.8565 33.8232 41.9236 54.8418 74.5401 99.9766 118.7169 1 1 2 . 6 4 87 88.4449 64.9101 48.4027 37 .8673  0.0 14.0705 31 .0458 56.8531 114.0342 529.9675 -530.0046 -114.0361 -56.8537 -31.0462 -14.0708 -0.0003 -12.1419 -1.9800 -0.0000 32.5362 32.5362 37.6848 42.7705 47.4545 -46.1662 -39.3745 -30.9503 -22 .4652 -14.3046 -6.4474 1.2560 8.9898 16.9319 25.2051 33.7424 41.8935 47.3642 44.1490 23.0642 -12 .9411 -40.0292 -47.7215 -44.0031 -36.2950 -27.7604 -19.3863 -11.3539 -3.5764 4.1144 11 . 9 0 3 9 19.9575 28.3506 36.8684 44.4270 47.6270 38.7395 10.3664 -25.2132 -44.8267 -47.1334 -41.3435 -33.1269  . . . . . Z S. . . . 0.0 90.0000 14.07 89.9919 31.05 89.9911 56.85 89.9890 114.03 8 9 . 9 8 28 529.97 89.9293 530.00 -89.9293 114.04 -89.9820 56.85 -89.9860 31.05 -89.9825 14.07 -89.9645 0.01 -1.6310 1 2 . 14 -89.9586 1.98 -89.7246 34.27 -0.0000 47.26 43.5095 47.26 43.5095 5 4 . 19 44.0638 6 2 . 13 43.5029 71.22 41.7859 75.68 -37.5900 5 9.93 -41.0699 47.36 -40.8105 37.54 -36.7611 30.25 -28.2224 25.72 -14.5174 24.51 2.9372 26.87 19.5477 32.35 31.5550 40.47 38.5184 5 1 . 15 41.2743 64.72 40.3366 81.52 35.5215 100.53 26.0508 116.29 11.4400 118.84 -6.2517 106.03 -22.1802 87 . 13 -33.2081 69.44 -39.3242 54.91 -41.3747 43.41 -39.7522 34.54 -34.1430 28.23 -23.7171 24.87 -8.2688 2.5.00 9.4743 28.58 24.6104 35.09 34.6682 4 4 . 13 39.9698 55.83 41.3290 70.58 39.0107 88.46 32.5764 107.22 2 1 . 1 8 06 119.17 4.9904 115.44 - 12.6161 99.16 -26.8773 80.22 -35.9346 63.66 -40.5025 50.31 -41.1799  . . -VPHASF 0.0 359.9922 359.9927 359.9932 359.9941 359.9956 180.1368 180.1393 180.1433 180.1509 • 180.1723 268.5090 180.1786 180.4150 180.1384 223.6479 223.6479 227.9999 231.7948 235.1664 235.1664 241.4944 249.3212 259.5381 273.3928 291.9338 314.0225 33 5 . 3 0 6 2 352.2737 4.7859 14.0980 21.3537 27.3297 32.5333 37.3137 41.9384 46.6464 51.6918 57.3925 64.1983 72.7912 84.2019 99.7310 119.9133 142.2797 162.1674 177.3666 188.5362 196.9739 203.6833 209.3262 214.3400 219.0363 223.6666 228.4704 233.7194 239.7710 247.1472  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  31 .3212 27.3793 25.2398 24.4938 25 .0047 26.8655 30.4273 36.4058 46 .0620 61.2986 83 .6280 108.8213 119.6688 104.5647 78.9972 57.9606 43.9200 35.0730 29.6199 26.4160 24.8251 24.5495 25 .5388 27.9748 32 .3267 39.4885 50.9676 68.7723 93.2572 115.6483 116.6405 95.1337 7 0 . 3 3 54 52.0142 40.1510 32.7395 28.2236 25.6716 24.5895 24.7777 26.2702 29.3448 34.6081 43.1599 56.7504 77.2436 102 .7482 119.2751 23.3254 18.5523 15 .6618 13.9225 12.9669 12.6171 12.8123 13.5860 15 .0782 15.0782 15 .0782 50.5991  -24.6034 -16.3593 -8.4408 -0.7191 6 .9849 14.8533 23.0296 31.5171 39.8772 46.3981 46.2443 30.4189 -3 .4780 -34.9242 -47.1571 -45.4914 -38.3915 -29.9351 - 2 1 .4927 -13.3811 -5 . 5 5 6 9 2.1342 9.8766 17.8444 26.1458 34.6726 42.6555 47.4937 42.6626 19.2946 -16.8842 -41.6845 -47.5814 -43.1604 -35.2835 -26.7581 -18.4335 -10.4450 -2.6934 4.9930 12.7985 20.8809 29.2954 37.7645 45.0319 47.2949 36.3836 6.1315 -47.3859 -35.6895 -25.7505 -16.9549 -8.8539 -1.0974 6.6173 14.5868 23.1429 23.1429 23.1429 -0.0001  - 238  39.83 31.89 26.61 24.50 25 .96 30.70 3 8 . 16 4 8 . 15 60.93 76.88 95 .56 112.99 119.72 110.24 9 2 . 00 73.68 58 .33 4 6 . 11 36.60 29.61 25 .44 24.64 27.38 3 3 . 18 41.58 52.55 66 .46 83.58 102.55 117.25 117.86 103.87 84.92 67.59 53.45 42.28 33.71 27.72 24.74 25.28 29.22 36.02 45 .34 57.35 72.45 90.57 109.00 119.43 52.82 40.22 3 0 . 14 21.94 15.70 12.66 14.42 19.93 27.62 27.62 27.62 50.60  -38.1503 -30.8585 -18.4911 -1.6815 15.6074 28.9372 37.1210 40.8832 40.8836 37.1228 28.9416 15.6173 -1.6648 -18.4691 -30.8349 -38.1272 -41.1575 -40.4810 -35.9653 -26.8645 -12.6172 4.9685 21.1429 32.5327 38.9658 41.2845 39.9264 34.6287 24.5828 9.4719 -8.2366 -23.6615 -34.0780 -39.6852 -41.3080 -39.2592 -33.1495 -22.1400 -6.2510 11.3933 25.9748 35.4346 40.2476 41.1856 38.4323 31.4784 19.4993 2.9428 -63.7915 -62.5334 -58.6913 -50.6089 -34.3257 -4.9708 27.3153 47.0344 56.9146 56.9146 56.9146 -0.0001  256.6477 269.4387 286.7412 308.2178 330.1521 348.3474 1.8997 11.9149 19.6165 25.8698 31.2389 36.1051 40.7516 45.4200 50.3567 55.8580 62.3313 70.3869 80.9573 95.3102 114.3647 136.5842 15 7.45 86 173.8667 185.9473 194.9302 202.0645 207.9380 213.0850 217.8423 222.4717 227.2121 232.3220 238.1307 245.1089 253.9716 265.7900 281.8535 302.4585 324.7358 344.1067 358.7795 9.5788 17.7827 24.3497 29.9094 34.8800 39.5639 39.5641 44.1116 50.6486 61.0653 79.4725 110.8494 145.1452 166.9498 179.0926  0.0  0.0 0.0  TRAP MAIN  - 239 T RI TUNE 3 DATE 01-05-72 CYCLOTRON 80 S E C T I O N S IN 2 •RESONATORS*  19:44:06 DEES  RESONANT FREQUENCY 23.1000 CHARACTERISTIC IMPEDANCE 46.00 T I P TO T I P C A P A C I T A N C E 6.50 V O L T A G E , C U R R E N T P E A K S , POWER. L O S S RMS AT S H O R T X = 0 V O L T A G E TO C U R R E N T P H A S E 9 0 HOT A R M L E N G T H = 3.06780 T I P TO T I P D I S T A N C E = 0.1524 A V E R A G E W I D T H OF S E C T I O N = 0.8320 M A X . V O L T A G E ON R E S # 1 = 99998.8 M A X . V O L T A G E ON R E S # 2 = 100005.7 VOLTAGE PHASE S H I F T = 179.86 MAX.CURRENT IN RES#1 = 2182.1 MAX.CURRENT IN RES#2 = 2182.2 C U R R E N T D E N S I T Y AT ROOT = 26.227 POWER L O S S D U E TO B E A M = 0. POWER L O S S I N R E S O N A T O R S = 959602. POWER L O S S I N R E S # 1 = 479595. POWER.LOSS INRES#2 = 480007. *COUPLING LOOP* VOLTAGE INDUCED IN LOOP = 9291.0 LOOP D I M E N S I O N S ARE H I G H T = 1.75 LENGTH = 17.50 WIDTH = 3.00 THICKNESS = 0.50 LOOP P O S I T I O N = 3.50 LOOP S E L F INDUCTANCE = 0.224169 CURRENT THROUGH LOOP = 206.57 POWER L O S S I N L O O P = 536.9 '-TRANSMISSION L I N E * C H A R . I M P E D A N C E OF L I N E = 54.17 OUTER D I A M E T E R = 11.100 INNER DIAMETER = 4.500 MAX.VOLTAGE WITHOUT C P ( 3 ) = 15928. MAX.CURRENT WITHOUT C P ( 3 ) = 294. MAX.VOLTAGE WITH C P ( 3 ) = 23285. MAX.CURRENT WITH C P ( 3 ) = 430. CURRENT THROUGH C P ( 2 ) = 228.99 C A P A C I T O R A F T E R LOOP C P ( 2 ) = 120.000 P O S I T I O N OF C P ( 2 ) = 0.610 CURRENT THROUGH C P ( 3 ) = 276. CAPACITOR C P ( 3 ) = 120.000 P O S I T I O N OF CP(3) = 31.9200 LENGTH AFTER C P ( 3 ) = 2.4800 T O T A L L E N G T H OF L I N E = 34.4000 VSWR-WITHOUT C P ( 2 ) = 1.96 VSW R W I T H O U T C P ( 3 ) = 2.44 VSWR A F T E R C P ( 3 ) = 5.19 POWER L O S S I N L I N E = 4979. POWER L O S S I N N E R C . = 3543. POWER L O S S O U T E R C . = 1436. •POWER T U B E * TUBE CURRENT = 170.59 TUBE VOLTAGE = 11311.8 C U R R E N T TO V O L T A G E P H A S E = -0.58 CAPACITOR C P ( 6 ) AFTER LINE = 208.994 CURRENT THROUGH C P ( 6 ) = 343.1 T O T A L POWER L O S S - - ' _ 964769.  MH Z OHMS  PFARAD DEGREES METERS METERS METERS  VOLTS VOLTS DEGREES AMPS AMPS AMPS/CM  WATTS WATTS WATTS WATTS VOLTS INCHES INCHES INCHES INCHES INCHES MICROHENRY AMPS  WATTS OHMS INCHES INCHES  VOLTS AMPS  VOLTS AMPS AMPS  PFARAD METERS AMPS  PFARAD METERS METERS METERS  WATTS WATTS WATTS AMPS VOLTS DEGREES PFARAD AMPS ' WATTS  - 240 •RESONATORS* AVERAGE MAGNETIC ENERGY 1 AVERAGE E L E C T R I C ENERGY 1 AVERAGE TOTAL ENERGY 1 AVERAGE MAGNETIC ENERGY 2 AVERAGE E L E C T R I C ENERGY 2 AVERAGE TOTAL ENERGY 2 Q U A L I T Y COMPUTEO QUALITY MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA FORESHORTENING ...CONDUCTIVITY SKIN DEPTH ' ALFA IN RESONATOR L U M P E D C A P A C I T A N C E AT V O l L U M P E D I N D U C T A N C E AT V O l LUMPED C A P A C I T A N C E AT V 0 2 LUMPED I N D U C T A N C E AT V 0 2 L U M P E D C A P A C I T A N C E AT 1 0 1 L U M P E D I N D U C T A N C E AT 101 L U M P E D C A P A C I T A N C E AT 1 0 2 LUMPED I N D U C T A N C E AT 102 EQU I V A L E N T V O L T A G E P E A K 1 EQUIVALENT VOLTAGE PEAK 2 ION O R B I T I N G FREQUENCY ACCELERATING TIME PERIOD TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN Q M E A S = SHUNT R E S I S T A N C E 1 AT V P E A K = SHUNT R E S I S T A N C E 2 AT V P E A K = SHUNT R E S I S T A N C E AT VLOOP= D E E TO D E E C A P A C I T A N C E AVERAGE ENERGY IN C P ( 1 ) T 0 T = R E A C T A N C E OF C T I P T O T A L D I S T R I B U T E D C A P A C I T Y DEE D I S T R I B U T E D I N D U C T A N C E DEE = RESISTANCE DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA I M A G . C O M P O N E N T OF Z R E S 2 * L 0 0 P AREA/RESONATOR AREA = RESONATOR S T E P U P R A T I O DEE S E R I E S R E S I S T A N C E POWER L O S S I N ROOT 1 P O W E R L O S S I N ROOT 2 MUTUAL I N D U C T A N C E AT LOOP = C O U P L I N G C O E F F . A T LOOP M U T U A L I N D U C T A N C E D I S T R I B . •= M U T U A L I N D U C T A N C E AT T U B E = COUPLING C O E F F . A T TUBE  11.84865 11.84862 23.69727 11.85037 11.35016 23.70052 7168.99 7168.99 7515.96 23.8549 12.98702 3.24675 145.14143 4.96 06 5.800 0.00137738 0.00003271 4739.59 0.0100156 4739.62 0.0100155 7630.28 0.0062212 7630.23 0.0062213 99999. 100006. 4.6 2 0 0 0.0 98.79 0.0 959602. 10425.20 10417.68 44.9783 260.00 2.6001081 -26.50 2898.551 0.0038333 0.0003009 0.1016 0.1016 0.39370 0.623377 0.4838048 -0.0031098 0.089647 10.76373 0.000 2028 14556.38 14558.39 0.0009305 0.01963754 0.0007 461 0.0011270 0.00312937  JOULES JOULES JOULES JOULES JOULES JOULES  MHZ METERS METERS 10**6RAD/SEC DEGREES 1 0 * * 7 MHOS CM 1/M PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY VOLTS VOLTS MHZ MICROSEC MICROSEC WATTS OHMS OHMS OHMS PFARAD JOULES OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS METER S**2 1/M OHMS  OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY  - 241 •COUPLING LOOP* 0.0037623 LOOP R E S I S T A N C E 32.53622 LOOP REACTANCE 0.019758 LOOP AREA . = 2 7 . 108 INPUT CURRENT D E N S I T Y 0.001290 LOOP S U R F A C E 0.0023913 AVERAGE ENERGY IN LOOP 186.94 L O S S D U E TO I N P U T C U R R E N T = 3 4 9 .96 L O S S DUE TO C A V I T Y C U R R E N T = •TRANSMISSION L I N E * I M A G . C O M P O N E N T OF Z T L -0.0050632 A L F A IN L I N E 0.00004522 DISTRIBUTED CAPACITY 61.5347 DISTRIBUTED INDUCTANCE 0.180567 R E S I S T A N C E PER UNIT LENGTH = 0.004899 OUTER RADIUS 0,1409701 INNER R A D I U S 0.0571500 AVERAGE ELECTRIC ENERGY 0.0941656 AVERAGE MAGNETIC ENERGY 0.0942017 AVERAGE TOTAL ENERGY 0. 1 8 8 3 6 7 3 TL Q U A L I T Y FACTOR 5490.77 LUMPED C A P A C I T A N C E AT VTUBE= 3.66592 LUMPED I N D U C T A N C E AT VTUBE = 12.94893 S E R I E S R E S I S T A N C E OF LINE = 0.3423 FIRST SECTION AVERAGE MAGNETIC ENERGY 0.0009742 AVERAGE ELECTRIC ENERGY 0.0014348 AVERAGE TOTAL ENERGY 0.0024090 POWER L O S S O U T E R C O N D U C T O R = 15.2 POWER L O S S I N N E R C O N D U C T O R = 37.6 POWER L O S S T O T A L 52.8 REFLECTION COEFFICIENT 0.324 R E A C T A N C E OF C P ( 2 ) -57.42 AVERAGE ENERGY IN C P ( 2 ) 0.0051858 SECOND S E C T I O N AVERAGE MAGNETIC ENERGY 0.0726283 AVERAGE ELECTRIC ENERGY 0.0700800 AVERAGE TOTAL ENERGY 0.1427084 POWER L O S S O U T E R C O N D U C T O R = 1136.1 POWER L O S S I N N E R C O N D U C T O R = 2802.4 POWER L O S S T O T A L 3 9 3 8 .6 REFLECTION COEFFICIENT 0.418 R E A C T A N C E OF C P ( 3 ) -57.42 INIT.VOLTAGE MAX.POSITION = 38.190 THIRD SECTION AVERAGE MAGNETIC ENERGY 0.0132080 AVERAGE ELECTRIC ENERGY 0.00 32 4 1 5 AVERAGE TOTAL ENERGY 0.0214494 POWER L O S S O U T E R C O N D U C T O R = 2 8 5 .0 POWER L O S S I N N E R C O N D U C T O R = 702.9 POWER L O S S T O T A L 987.9 REFLECTION COEFFICIENT 0.6 77 P O S I T I O N OF V O L T A G E M A X I M U M = 36.670 AVERAGE ENERGY IN C P ( 3 ) 0.0075 381 •POWER T U B E * R E S I S T A N C E TO M A T C H R E A C T A N C E TO MATCH R E A C T A N C E OF C P ( 3 ) R E S I S T A N C E OF T U B E AVERAGE ENERGY IN C P ( 6 )  66.3152 32.8017 -32.9667 66 . 3 0 8 4 0.0066856  OHMS OHMS " METERS* *2 AMPS/CM MET E R * * 2 JOULES WATTS WATTS :  OHMS 1/M . PFARAD/M MICROHENRY/M OHMS/M METERS METERS JOULES JOULES JOULES PFARAD MICROHENRY OHMS JOULES JOULES JOULES WATTS WATTS WATTS OHMS JOULES JOULES JOULES JOULES WATTS WATTS WATTS OHMS METERS JOULES JOULES JOULES WATTS WATTS WATTS METERS JOULES OHMS OHMS OHMS OHMS JOULES  S T N . .METERS., 1 0.0 2 0.6136 3 0.6136 4 0.6136 0.6136 5 6 0.6136 7 0.0 8 0.6136 9 0.6136 0.6136 10 11 0.6136 12 0.6136 13 0.0 14 0.0 15 0.0 16 0.0 17 0.0 18 0.2033 19 0.2033 20 0.2033 21 0 .0 22 0.3684 23 0.3684 24 0.3684 25 0.3684 26 0.3684 27 0.3684 28 0.3684 29 0 .3684 30 0.3684 31 0.3684 32 0.3684 33 0.3684 34 0.3684 35 0.3684 36 0.3684 37 0 .3684 38 0.3684 39 0.3684 40 0.3684 41 0 .3684 42 0.3684 43 0.3684 44 0.3684 45 0 .3684 46 0.3684 47 0.3684 48 0.3684 49 0.3684 50 0.3684 51 0.3684 52 0.3684 53 0.3684 54 0.3684 55 0.3684 56 0.3684 57 0.3684 58 0.3684  . . E d ,K ) . . , 0.0 29359.91 56151.70 7 8 0 3 1 .81 93086.44 9 9 9 9 8 .75 -100005.50 -93092.81 -78037.25 - 5 6 1 5 5 .87 -29362.35 -0 .55 -25618.55 -4316.61 -9290.97 -9276.54 -9276.54 -9229.22 -9092.66 - 8 8 6 8 .16 -8868.16 - 6 7 6 1 .58 -4440.77 -1979.25 545.02 3 0 5 2 .06 5462.45 7 6 9 9 .86 9693.39 11379.88 12705.91 13629.46 14121.27 14165.77 13761.51 1 2 9 2 1 .32 11671.77 1 0 0 5 2 .48 8114.72 5919.86 3537.43 1042 .91 -1484.69 - 3 9 6 5 .32 -6320.36 - 8 4 7 5 .23 -10361.66 -11919.88 -13100.54 -13866.21 -14192.64 -14069.48 -13500.62 -12504.08 -11111.43 -9366.75 -7325.33 - 5 0 5 1 .82 :  - 242 i E(2,K) ...ET(K).. 0.0 0.0 -3.91 29359.9 - 7 . 14 56151.7 -9.08 78031.8 -9.31 9 3 0 8 6 .4 -7.63 9 9 9 9 8 .8 -238.84 100005.7 -226.38 93093.1 -195.22 78037.4 -147.89 5 6 1 5 6 .0 -88.29 29362.5 -21 .31 21.3 -79.85 25618.7 -31.27 43 16.7 -22.45 9291.0 -6 7 4 3 . 3 2 11468.5 -6743.32 11468.5 -7809.71 12090. 1 -8800.61 12654.1 -9706.43 13147.6 -9706.43 .13 1 4 7 . 6 -9479.83 11644.1 -8953.01 9993.8 -8142.63 8379.7 -7074.34 7095.3 -5781.96 6538.1 -4306 .43 6955.8 -2694.46 8157.7 -997.09 9744.5 731.92 11403.4 2437.81 12937.6 4066.56 14223.2 5566.57 15178.8 6890.36 15752.6 7995.97 15915.9 8848.39 15660.6 9420.61 14999.3 96 9 4 . 5 1 13965.5 9661 .39 12617. 1 9322.28 11043. 1 8687.93 .9380.5 7778.41 7848.0 6622.52 6786.9 5256.84 65 84 3724.63 7336 2074.41 8725 358.43 103 6 7 -1368.96 11998 -3 05 3.07 13451 -4640.5 5 14622 -6081 . 11 15440.6 - 7 3 2 9 . 14 15864.0 - 8 3 4 5 . 11 15371.6 -9096.82 15463.0 -9560.46 14658.3 -9721.33 13499.6 -9574.34 12055.2 -9124.13 10429.3  I  ( 1 » K ) .. I.( 2 » K} . I.(.K ) . . . .  0.0 0.01 0.05 0. 10 0. 16 0. 22 0.22 1.73 3. 10 4.21 4. 98 5,.34 5.05 5.31 -206.57 -206.57 -206.57 -193.34 -178.25 -161.43 7.63 39.27 69.67 97 .87 122.97 1 4 4 . 17 160.81 172.35 178.44 178.88 173.65 162.93 147.04 126.49 101.94 7 4 . 16 44.03 12.50 -19.42 -50.73 -80.43 -107.59 -131.34 -150.94 -165.75 -175.31 -179.33 -177.66 -170.37 -157.68 -139.99 -117.88 -92.03 -63.26 -32.49 -0.69 3 1 . 14 61.98  -2182.06 -2086.63 -1808.67 -1372.52 -816.30 -188.69 -188.69 -816.35 -1372.60 -1808.79 -2086.76 -2182.20 -2109.94 -2180.19 - 0 . 50 - 0 . 50 -0.50 -17.32 -33.97 -50.29 -204.74 -230.52 -249.00 -259.59 -261.95 -256.02 -241.98 -220.27 -191.58 -156.83 -117.10 -73.67 -27.90 18.76 6 4 . 82 108.82 149.39 185.21 215.18 238.32 253.92 261.48 260.75 251.76 234.80 210.39 179.33 142.58 101.31 56.84 10.56 -36.05 -81.53 -124.41 -163.36 -197.14 -224.67 -245.08  2182. 2087. 1809. 1373. 816. 189. 189. 316. 1373. 1309. 2087. 2182. 2110. 2180. 207. 207. 207. 194. 181. 169. 205. 234. 259. 277. 289. 294. 291. 280. 262. 238. 209. 179.  150.  128. 121. 132. 156. 186. 216. 244. 266. 283. 292. 294. 287. 274. 254. 228. 198. 168. 140. 123. • 123. 140. 167. 197. 227. 253.  P(  l K ) . . .P(2,K) 0.0 0.0 8536.86 0.0 7119.47 0.0 4769.78 0.0 2 2 9 1 .92 0.0 533.90 0.0 0.0 0.0 533.94 0.0 2292.20 0.0 4770.39 0.0 7120.43 0.0 8538.01 0.0 0.0 0.0 0.0 0.0 186.94 0.0 0.0 0.0 0.0 0.0 20.00 0.0 17.57 0.0 15.30 0.0 0.0 0.0 43.58 0.0 54.93 0.0 65.09 0.0 72 .79 0.0 77.06 0.0 77.37 0.0 73.67 0.0 66 .43 0.0 56.57 0.0 45.32 0.0 34.09 0.0 24.30 0.0 17.17 0.0 13.61 0.0 14.06 0.0 18.47 0.0 26.28 0.0 36 .50 0.0 47.87 0.0 58.93 0.0 68.31 0.0 74.83 0.0 77.67 0.0 76 .47 0.0 71.37 0.0 63 .04 0.0 52.50 0.0 41 .08 0.0 30.23 0.0 21 .30 0.0 15.41 0.0 13.32 0.0 15.27 0.0 21.03 0.0 29.88 0.0 40.69 0.0 52.11 0.0  - 243 -  t  0. 8537. 7119. 4770. 2292. 534. 0. 534. 2292. 4770. 7120. 8538. 0. 0. 187. 0. 0. 20. 18. 15. 0. 44. 55. 65. 73. 77. 77. 74. 66. 57. 45. 34. 24. 17. 14. 14. 18. 26. 37. 48. 59. 68. 75. 78. 76. 71 . 63. 52. 41. 30. 21. 15. 13. 15. 21. 30. 41. 52.  0. 8537. 15656. 20426. 22718. 23252. 23252. 23786. 26078. 30848. 37969. 46507. 46507. 959602. 959789. 959789. 959789. 959809. 959827. 959842. 959842. 959886. 959941. 960006. 960078. 960155. 960233. 960306. 960373. 960429. 960475. 960509. 960533. 960550. 960564. 960578. 960596. 960622. 960659. 960707. 960766. 960834. 960909. 960986. 961063. 961134. 961197. 961249. 961291. 961321 . 961342. 961357. 961371. 961386. 961407. 961437. 961477. 961529.  • • •Z P R• • » * 0.0 100974.3 201388.9 298096.8  0.0 14.1 31.0 56.9  1 2 3 4  381418.3 114.0 430059.4 530.0 430119.2 -530.0 364345.4 -114.0 -56.9 233519.4 102226.0 -31.0 22706.9 -14.1 0.0 -0.3 16815.1 -12.1 411.9 -2.0 j ; si, <J. ~t1- 1- - r T4 5 . 0 * *i*si. ,u 68.5 94.7 68.5 94.7 76.1 108.3 127.1 83.4 90.0 154.2 90.0 -91.5 70.6 -70.2 52.0 -57 .7 36.6 -53.6 26.2 -69.2 -818.9 22.3 76.9 25.2 34.6 54.0 49.4 56.5 67.7 67.9 87. 1 87.6 105.3 121.4 119.9 190.0 129.2 409.6 131.9 -3295.9 127.7 -327.0 117. 1 -169.3 -112.0 101.5 82.9 -82.3 63 .5 -64.7 45.8 -55.1 32. 1 -55.5 24.0 -95.4 22 .6 211.6 28.0 62.1 39.6 53.6 55.9 59.9 74.9 74. 1 94. 1 97 .9 111 .2 140.7 238. 1 124.0 130.9 705.6 131.0 -757.2 -243.9 124.4 111.7 -142.8 94.8 -99.1 75.6 -74. 8 56.6 -60.3  5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118  0.3684 -2618.25 0 .3684 - 1 0 1 .68 0.3684 2 4 1 8 .15 0 .3684 4861.42 0.3684 7 1 5 0 .73 0.3684 9213.55 0.3683 1 0 9 8 4 .49 0 .3684 12407.60 0.3683 1 3 4 3 7 .62 0.3684 14042.02 0.3683 . 14201.58 0 .3683 13911.28 0.3684 1 3 1 8 0 .27 0.3683 12031.75 0.3683 1 0 5 0 2 .08 0 .3684 8639.63 0.3683 6 5 0 3 .54 0.3683 4 1 6 1 .39 0.3684 1687 .29 0 .3683 -840.21 0.3684 -3341.25 0 .3683 -5736.36 0.3683 -7949.82 0.3684 -9911.57 0.3683 -11559.29 0.3683 -12840.86 0.3684 -13715.75 0 .3683 -14156.10 0.3683 - 1 4 1 4 8 .07 0 .3684 -13691.80 0.3683 - 1 2 8 0 1 .86 0 .3684 -11506.27 0.3683 -9846.25 0 .3683 -7874.29 0.3684 - 5 6 5 2 .77 0.3683 -3252.23 0.3683 - 7 4 8 .64 0 .3684 1778.82 0.3683 4 2 4 9 .89 0.3683 6 5 8 6 .34 0.3684 8714.31 0.3683 10566.16 0.3684 1 2 0 8 3 .40 0 .3683 13217.82 0.3683 1 3 9 3 3 .58 0 .3684 14207.95 0.3683 14032.26 0.3684 13412.01 0.0 1 3 4 1 1 .97 0.3100 10668.59 0.3100 7 6 8 5 .67 0.3100 4530.15 0.3100 1272 .89 0.3100 -2012.96 0.3100 - 5 2 5 3 .65 0.3100 -8376.37 0.3100 - 1 1 3 1 1 .05 0.0 -11311.05 0.0 - 1 1 3 1 1 .05 0.0 -11311.05  -8384.93 -7380.16 -6141.62 -4708.53 -3126.27 -1444.94 2 8 2 . 15 2000.47 3655.44 5194.75 6569.55 7736.35 8.65 8.2 1 9 3 0 5 .88 9658.85 9705.95 9445.66 8 8 8 6 .22 8045.31 6949.61 5633.71 4139.42 2513.97 808 .77 -922.02 -2623.69 -4242.40 -5726.73 -7029.77 -8110.26 -8933.91 -9474.66 -9715.36 - 9 6 4 8 .38 -9275.84 -8609.52 -7670.53 -6488.52 -5101.04 -3 5 5 1 . 9 6 -1890.26 -168.71 1558.32 3235.99 4811.25 6234.23 7459.80 8449.18 8449.21 7884.79 7 1 4 3 .39 6 2 4 1 .63 5199.76 4 0 4 1 . 16 2791.81 1479.77 134.46 134.46 134.46 134.46  244 8784.2 7380.9 6600.5 6767.8 7804.3 93 26 2 1 0 9 8 8 .1 1 2 5 6 7 ,8 1 3 9 2 5 ,9 14972. 1 15647.5 15917.7 15769.7 15210.6 14268.4 12994.2 11468.1 9812.3 8220.3 7000.2 6550.0 7073.9 8337.8 9944.5 11596.0 13106.2 14356.9 15270.6 15798.3 15913.6 15611.0 14905.1 13832.4 12453.7 10862.5 9203.3 7707.0 6727.9 6639.4 7483.1 8917.0 10567.5 12183.5 13608 14740 15515 15891 1585 1 15851 13266 10492.7 7712.3 5353.3 4514.7 5949.4 8506.1 11311.8 113 11.8 11311.8 11311.8  90.85 116.86 1 3 9 . 16 157.05 169.97 177.51 179.43 175.67 166.34 151.74 132.34 108.74 81.70 52.08 20.80 -11.14 -42.73 -72.96 -100.89 -125.62 -146.37 -162.49 -173.46 -178.94 -178.76 -172.91 -161.58 -145.14 -124.10 -99.13 -71.02 -40.66 -9.01 22.93 5 4 . 14 83.64 110.49 133.84 152.95 167.22 176.20 179.60 177.30 169.40 156.12 137.91 115.32 89.08 -5 8.08 -80.73 -101.58 -120.14 -136.01 -148.82 -158.30 -164.23 -166.47 0.0 170.58 170.58  -257.73 -262.22 -258.40 -246.39 -226.58 -199.59 -166.28 -127.70 -85.07 -39.75 6.83 53.20 97.88 139.46 176.63 208.20 233.17 250.76 260.41 261.81 254.91 239.94 217.37 187.91 152.49 112.25 68.45 22.48 -24.20 -70.11 -113.81 -153.90 -189.12 -218.34 -240.65 -255.34 -261.94 -260.24 -250.30 -232.43 -207.19 -175.39 -138.03 -96.31 -51.53 - 5 . 11 41.46 86.73 320.33 353.72 379.18 396.12 404.17 40 3.15 39 3.07 3 7 4 . 1,6 346.86 -344.86 0.0 -1.73  273. 287. 293. 292. 283. 267. 245. 217. 187. 157. 133. 121. 127. 149. 178. 208. 237. 261. 279. 290. 294. 290. 278. 259. 235. 206. 175. 147. 126. 121. 134. 159. 189. 220. 247. 269. 284. 293. 293. 286. 272. 251. 225. 195. 164. 138. 123. 124. 326. 363. 393. 414. 426. 430. 424. 409. 385. 345. 171. 171.  62.70 71.14 76.37 77.72 75.02 68.63 59.33 48.30 36.93 26.64 18.73 14.19 13.59 17.02 24.03 33.75 44.95 56.23 66.17 73 .52 77.36 77.20 73.06 65 .48 55.38 44 .06 32.93 23.38 16.62 13.51 14.42 19.26 27.40 37.83 49.23 60 .17 69.28 75 .41 77.79 76 .13 70.63 61 .97 51 .27 39.84 29.14 20 .51 15.03 13.39• 0.0 90.35 108.76 123 .95 134.57 139 .67 138.79 132.02 119.95 0.0 0.0 0.0  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 . 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  63. 71. 76. 78. 75. 69. 59. 48. 37. 27. 19. 14. 14. 17. 24. 34. 45. 56. 66 . 74. 77. 77. 73. 65. 55. 44. 33. 23. 17. 14. 14. 19. 27. 38. 49. 60. 69. 75. 78. 76. 71. 62. 51 . 40. 29. 21. 15. 13. 0. 90. 109. 124. 135. 140. 139. 132. 120. 0. 0. 0.  - 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