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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., Technical U n i v e r s i t y of Prague, Czechoslovakia, 1967 A thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements for the degree of Doctor of Philosophy i n the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1972 In presenting th i s thesis i n p a r t i a l fulf i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library s h a l l make i t f reely avai lable for reference and study. I further agree that permission for extensive copying of th i s thesis for scholar ly purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or publ ica t ion of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permission. Department of P h y s i c s The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada D a t e March 1, 1972 ABSTRACT The basic design of the resonator system, coupling loop assembly and the resonant transmission l i n e for the TRIUMF isochronous cyclotron i s studied i n d e t a i l and the beam-resonator RF f i e l d i n t e r a c t i o n i s estimated using a simple model. The matching techniques which w i l l allow a maximum power transfer from the power tube stage to the Dees are discussed. Two major computer programs have been written and are used to determine the parameters of the RF system necessary i n order to obtain a desired impedance match and to f i n d the various resonant frequencies of the RF system. The r e s u l t s of a large number of experimental tests c a r r i e d out i n order to v e r i f y the resonator 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 investigate the resonator frequency tuning and the resonator mechanical construction around the centre post, and to apply the proposed matching technique to the TRIUMF RF system, are given. The r e s u l t s from the prototype resonator segments and the transmission l i n e tested at high power both i n a i r and under high vacuum indic a t e that the RF system performance i s compatible with the cyclotron operation. i i C O N T E N T S Page I. INTRODUCTION 1 1. TRIUMF Cyclotron i n General 1 2. TRIUMF Cyclotron RF System 3 3. C r i t i c a l RF Parameters 5 I I . THEORETICAL CALCULATIONS 9 1. Resonator Basic Parameters and Relations 9 1.1 C h a r a c t e r i s t i c Impedance 9 1.2 Tip Loading Capacity 10 1.3 Resonant Frequency 11 1.4 Resonator Power Loss 12 1.5 Energy Stored i n Resonator 14 1.6 Resonator Quality Factor 15 2. RF Tolerances 18 2.1 Frequency Detuning due to Evacuation of the Vacuum Chamber 18 2.2 Detuning of Resonator due to Temperature V a r i a t i o n 19 2.3 Voltage V a r i a t i o n due to Temperature V a r i a t i o n 21 2.4 Frequency Detuning due to E l e c t r o s t a t i c and Magnetic Forces 22 2.5 Tolerances to be Met 25 3. Voltage Breakdown 27 3.1 Sparking 27 3.2 Multipactoring Ranges 29 3.3 Rate of Rise of Resonator Voltage 32 4. Frequency Tuning 40 4.1 Root Shorting Plane Motion 40 4.2 Capacitive E f f e c t Near the Accelerating Gap 40 4.3 Ground Arm Tip D e f l e c t i o n 41 4.4 Tuning Bellows at the Root 42 4.5 Third Harmonic Tuning Diaphragms 44 5. Resonator Modifications 49 5.1 Extreme End Segments 49 5.2 Central Region Segments 50 6. Beam Loading 54 6.1 Beam-RF F i e l d Interaction 54 6.2 Beam Induced Voltage 61 6.3 Conclusive Remarks 64 7. Coupling Loop Assembly 68 7.1 Power Loss i n the Vacuum Seal 68 7.2 Power Loss i n Adjacent Areas 69 7.3 Power Loss i n the Coupling Loop 70 i i i Page 8. Transmission Line 72 9. Lumped Parameter Representation 75 9.1 Resonator Lumped Constants 75 9.2 Representation of the Coupling 76 9.3 Lumped Constant Representation of a Resonant Line 80 9.4 Representation of the Whole System i n Terms of Lumped Parameters 82 10. Resonant Operation of the RF System 83 10.1 Operation at the Maximum Power Transfer 83 10.2 Other Resonances of the System 88 10.3 Possible Operating Conditions 91 I I I . EXPERIMENTAL TESTS 95 1. Measurement of Resonator Parameters 97 2. Frequency Tuning 99 2.1 Tuning Stub 99 2.2 Ground Arm Tip D e f l e c t i o n 100 2.3 C y l i n d r i c a l Capacitors 101 2.4 Capacitive Plates 102 2.5 Inductive Loops at the Root 102 2.6 Third Harmonic Tuning Diaphragms 103 3. Voltage and Frequency Variations due to Mechanical Misalignments 107 4. Resonator Modifications 108 4.1 Extreme End Segments 108 4.2 Central Region Segments 108 5. Resonant Line as a Matching Network 111 IV. CENTRAL REGION CYCLOTRON 117 1. Low Power Level 117 1.1 Quality Factors 117 1.2 Resonator Frequencies 118 1.3 Fine Frequency Tuning 119 1.4 Coarse Frequency Tuning 120 1.5 RF Probes 122 1.6 Coupling Loop 123 1.7 Transmission Line and Resonances of the System 124 1.8 Comparison of Results from Lumped and D i s t r i b u t e d Parameter Representations 127 2. High Power Level 129 2.1 Sparking and Multipactoring 129 2.2 RF Contacts 131 2.3 Voltage and Frequency S t a b i l i t y 131 2.4 Coupling Loop and Transmission Line 133 2.5 RF Power Amplifier 134 iv Page V. SUMMARY AND CONCLUSIONS 135 References 137 Figures 138 Appendix A. Centre Region Cyclotron RF System Parameters -the RF Fundamental 209 Appendix B. Centre Region Cyclotron RF System Parameters -the RF Third Harmonic 222 Appendix C. Main Cyclotron RF System Parameters - the RF Fundamental 229 v LIST OF TABLES Page I. Tolerances on the RF parameters 8 I I . Beam c h a r a c t e r i s t i c s 8 I I I . Percentage resonator frequency change vs temperature changes of resonator panels 20 IV. Resonator voltage v a r i a t i o n caused by temperature v a r i a t i o n 22 V. Permissible l e f t - r i g h t voltage asymmetry 26 VI. Permissible top-bottom voltage asymmetry 27 VII. K i l p a t r i c k ' s c r i t e r i o n applied to TRIUMF resonator geometry 29 VIII. Threshold multipactoring voltages for the RF fundamental 31 IX. Threshold multipactoring voltages f o r the t h i r d harmonic 31 X. Percentage frequency change vs p o s i t i o n of tuning bellows 43 XI. Calculated gaps and power loss f o r a modified extreme end segment 50 XII. Resonator power loss f o r a given RF voltage amplitude 60 XIII. Computed and measured frequency s h i f t s caused by turning the diaphragms 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 106 XIV. Calculated and measured dimensions of the f u l l - s c a l e s i n g l e resonant section and the resonant l i n e 112 XV. Calculated 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 section and the resonant l i n e 113 XVI. Quality f a c t o r and power absorbed vs tube simulating resistance 114 XVII. Measured and computed q u a l i t y factors of the CR cyclotron 117 XVIII. Measured frequency s h i f t s caused by tuning bellows 121 XIX. Measured frequency s h i f t s caused by ground arm t i p d e f l e c t i o n 121 v i 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 v i i LIST OF FIGURES Page 19. Schematic of the tuning bellows 156 20. Percentage frequency change vs p o s i t i o n of tuning bellows (CRM) a) the fundamental b) the t h i r d harmonic; cal c u l a t i o n based on perturbation theory by Slater 157 21. Percentage frequency change vs p o s i t i o n of diaphragm i n the resonator calculated using perturbation theory by Sla t e r (zero p o s i t i o n refers to the root ; 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 ) 158 22. Percentage frequency change vs p o s i t i o n of diaphragm i n the resonator; c a l c u l a t i o n based on a change of the t o t a l capacitance of the resonator (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 ) 159 23. Extreme end section a) top view b) section with modified hot arms 160 24. Central resonator segments a) top view b) side view c) g r i d of points f o r voltage measurements 161 25. Beam-RF f i e l d i n t e r a c t i o n a) resonator fed by an RF generator and a beam current generator b) amplitude of a beam pulse c) representation of a resonator with a beam load d) beam induced voltage and t o t a l voltage on resonator 162 26. Power delivered to 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) 163 B B 27. 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 yA, E^ = 500 MeV) 164 B 28. Power delivered to the beam a) and frequency change b) vs i n j e c t i o n phase (the RF fundamental, I = 750 yA, E c = 400 MeV) 165 29. 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 = 750 yA, E,, = 400 MeV) 166 B 30. Power delivered to the beam vs beam phase width a) the RF fundamental b) the t h i r d harmonic ( I = 100 yA, E_ = 500 MeV) B 167 B 31. Frequency change vs beam phase width (1^ =100 yA, Eg = 500 MeV) a) the RF fundamental b) the t h i r d harmonic 168 32. T o t a l RF fundamental voltage on resonator during the i n j e c t i o n of a beam (without the RF t h i r d harmonic) 169 33. T o t a l RF fundamental voltage on resonator during the i n j e c t i o n of a beam (with the RF t h i r d harmonic) 170 ix Page 34. T o t a l RF t h i r d harmonic voltage on resonator during the i n j e c t i o n of a beam (with the RF t h i r d harmonic) 171 35. Components of resonator voltage during the course of acce l e r a t i o n a) RF voltage amplitudes due to external sources b) the f i r s t harmonic component of beam induced voltage (RF fundamental operation) 172 36. Components of resonator voltage during the course of acceleration (RF f l a t - t o p operation) a) the f i r s t harmonic component of beam induced voltage b) the t h i r d harmonic component of beam induced voltage 173 37. Coupling loop assembly 174 38. Representation of coupling a) and lumped parameter representation of a Dee resonant system b) 175 39. Input impedance vs detuning from resonance a) magnitude of input impedance b) phase of input impedance 176 40. Resonator peak voltage vs detuning from resonance 177 41. Representation of the system with d i s t r i b u t e d parameters (program RESLINE) 178 42. Matching using a nA/2 l i n e a) representation of a Dee with 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 f Q c) impedance transformation using TT network 179 43. Representation of the system a)two Dees represented with lumped parameters and the transmission l i n e with d i s t r i b u t e d parameters (program MATCH) b) lumped parameter representation of the whole system 180 44. Phase s h i f t between loop and resonator root currents vs dr i v i n g frequency 181 45. Resonator t i p to loop voltage r a t i o vs d r i v i n g frequency a) magnitude of V,/V" b) phase of V,/ V" 46. Percentage frequency change vs stub shorting plunger 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 of 10 sections b) 4 stubs coupled c a p a c i t i v e l y to a two Dee resonator, each Dee made up of 10 sections 183 47. Voltage v a r i a t i o n along the accelerating gap vs stub shorting plunger 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 of 10 sections b) 4 stubs coupled c a p a c i t i v e l y to a two Dee resonator, each Dee made up of 10 sections 184 48. Percentage frequency change and qu a l i t y factor v a r i a t i o n vs stub shorting plunger 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 of 5 sections) 185 x Voltage 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 of 5 sections) a) the fundamental b) the t h i r d harmonic Frequency tuning by means of c y l i n d r i c a l capacitors (Dee made up of 10 sections) a) percentage frequency change and b ) q u a l i t y factor v a r i a t i o n vs p o s i t i o n of c y l i n d r i c a l capacitors Voltage v a r i a t i o n along the hot arm t i p s vs p o s i t i o n of c y l i n d r i c a l capacitors Quality factor v a r i a t i o n a) and voltage v a r i a t i o n along the hot arm t i p s b) vs p o s i t i o n of capacitive tuning plates (Dee made up of 10 sections) Frequency tuning by means of ro t a t i n g f i n s (measurements done with a 3 section resonator with the f i n s inserted i n the upper and lower centre segments) a) percentage frequency change and b) q u a l i t y factor v a r i a t i o n vs p o s i t i o n of tuning f i n s Frequency tuning by means of r o t a t i n g f i n s (measurements done with a two section resonator with 8 f i n s per 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 tuning by means of rot a t i n g loops (measurements done with a two section resonator with 8 loops per 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 tuning by means of diaphragms a) schematic of the experimental arrangement b) resonator with tuning diaphragms Frequency tuning by means of tuning 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 of diaphragms; p o s i t i o n of diaphragms i n resonator at 108.5 i n . from root Frequency tuning by means of tuning 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 of diaphragms; p o s i t i o n of diaphragms i n resonator at 65.4 i n . and 21.65 i n . from root Resonator frequency tuning by means of a tuning stub a) experimental arrangement b) lumped constant representation Percentage voltage 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 of the upper hot arm #3 i n an 18 section one Dee resonator b) d e f l e c t i o n of the lower hot arm #6B i n a 20 section two Dee resonator 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 of the upper hot arm #3 i n an 18 section one Dee resonator b) d e f l e c t i o n of the lower hot arm #6B i n a 20 section two Dee resonator 198 62. Percentage voltage change vs root plunger p o s i t i o n a) motion of root plungers i n section #7 i n ah 18 section one Dee resonator b) motion of root plungers i n section #7A i n a 20 section two Dee resonator 199 63. Percentage frequency change vs root plunger p o s i t i o n a) motion of root plungers i n section #7 i n an 18 section one Dee resonator b) motion of root plungers i n section #7A i n a 20 section two Dee resonator 200 64. Percentage frequency change a) and q u a l i t y factor v a r i a t i o n b) vs motion of root plungers i n section #4 i n a 5 section one Dee resonator 201 65. Central region geometry 202 66. Voltage d i s t r i b u t i o n along the transmission l i n e ; basic set-up 203 67. Voltage d i s t r i b u t i o n along the transmission l i n e ; magnitude of CP(3) changed to 415 pF 204 68. Voltage d i s t r i b u t i o n along the transmission l i n e ; p o s i t i o n of CP(3) changed to 327 cm 205 69. Voltage phase along the transmission l i n e ; basic set-up 206 70. CRM one Dee resonator 207 71. RF contacts 208 xxx ACKNOWLEDGEMENTS I would l i k e to thank Dr. K.L. Erdman for supervising t h i s work and for providing guidance and h e l p f u l suggestions throughout the course of my studies at U.B.C. I would also l i k e to thank Mr. O.K. Fredriksson f o r many h e l p f u l discus-sions and the members of 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 wife J i t k a f o r drawing the fi g u r e s . F i n a n c i a l support from the TRIUMF project throughout the course of t h i s work i s g r a t e f u l l y acknowledged. x i i i CHAPTER I. INTRODUCTION 1. TRIUMF CYCLOTRON IN GENERAL Tri- 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 isochronous sector-focused negative ion c y c l o t r o n , 1 designed to accelerate 100 yA of H ions to an energy of 500 MeV. Azimuthally-varying magnetic f i e l d i s provided by a s i x - s e c t o r 2 — magnet. The l i f e t i m e of H ions requires that the maximum magnetic f i e l d must be as low as 5.76 kG at 500 MeV radius to prevent excessive d i s s o c i a t i o n of H ions. As a consequence of a low maximum magnetic f i e l d , the 500 MeV o r b i t radius i s approximately equal to 308 i n . and the ion r o t a t i o n frequency i s only 4.62 MHz. The beam of H ions i s produced externally i n a hot filament Penning arc source of the Ehler's type. The beam i s accelerated to 300 keV and i s inj e c t e d a x i a l l y into the median plane of the cyclotron by a s p i r a l e l e c t r o -. r-, 3 s t a t i c m f l e c t o r . The RF system consists of two 180 deg wide Dees operating i n the push-pull mode. The RF frequency i s the f i f t h harmonic of the ion r o t a t i o n frequency. Since the Dee voltage i s 100 kV, the p a r t i c l e s gain 400 keV on each turn. The beam must thus complete 1250 turns before i t reaches 500 MeV. The RF transmitter must be capable of d e l i v e r i n g up to 1.5 MW (cw) of RF power dissip a t e d i n the r e s i s t i v e losses and the power supplied to the beam. To keep the r a d i a t i o n l e v e l below permissible l i m i t s a very good vacuum - 7 of the order of 10 Torr must e x i s t . E x traction i s achieved by s t r i p p i n g both electrons on the ions by passing the beam through a t h i n f o i l , thus reversing the curvature of the o r b i t i n the magnetic f i e l d . 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 at d i f f e r e n t distances from the centre of the cyclotron, the output energy varies between it 200 MeV and 500 MeV. The beam i s thus extracted with nearly 100% e f f i c i e n c y . Another advantage of t h i s extraction mechanism i s a p o s s i b i l i t y of simul-- 2 -taneous extraction of several proton beams at d i f f e r e n t energies. The cyclotron w i l l produce mesons which can be used i n examining the structure of the mesons themselves, i n photographing atomic n u c l e i with meson beams, and i n forming muonic and p i o n i c atoms. Two basic modes 5 of cyclotron operation are envisaged. Since the mesons are produced i n secondary targets the current must be as large as 200 uA, but the energy r e s o l u t i o n w i l l be low, AE = ±600 keV. This mode of operation w i l l require as high a duty cycle as possible. A beam with a high energy r e s o l u t i o n of AE = ±50 keV w i l l be produced during the other mode of cyclotron operation, i . e . separated turn a c c e l e r a t i o n . In t h i s case, the microscopic duty factor w i l l be very low and, consequently, the beam current w i l l be only 20 yA. Adding the t h i r d harmonic 5 of the RF to the RF fundamental f l a t - t o p s the RF voltage wave and helps to e i t h e r increase the phase acceptance giving a la r g e r beam current or to maintain s p a t i a l turn separation out to ex t r a c t i o n . - 3 -2. TRIUMF CYCLOTRON RF SYSTEM The TRIUMF RF system possesses unique features that have not been applied anywhere e l s e . The resonator i s made up of two Dees subtending an angle of 180 deg at the accelerating gap. The Dee structure i t s e l f (Fig. 1) consists of two quarter-wave long,shorted transmission l i n e s coupled c a p a c i t i v e l y at the high voltage end, which r e s u l t s i n two modes of possible operation. Only one of them, namely a push-pull mode, i s useful f o r acceleration of ions. The resonator i s divided into segments ( F i g . 2) i n order to s i m p l i f y manufacturing and handling and to avoid p a r a s i t i c modes. Each Dee i s made up of two rows (upper and lower), each of them cons i s t i n g of 20 resonator segments (Fig.3). The resonator panels are cantilevered, and thus the troubles frequently experienced i f i n s u l a t o r s are used can be avoided. In the TRIUMF cyclotron the beam of 100 yA of H~ ions w i l l be accelerated to the output energy of 500 MeV. The e l e c t r i c d i s s o c i a t i o n 7 of H ions i n a magnetic f i e l d l i m i t s the maximum magnetic f i e l d at the 500 MeV o r b i t to about 5.76 kG. A higher f i e l d would bring about higher beam losses and an unacceptable r a d i a t i o n . This l i m i t on the magnetic f i e l d places the ion o r b i t i n g frequency at approximately 4.62 MHz, which i s rather a low frequency as compared to other cyclotrons. To reduce the s i z e of the Dees and the power required the RF system w i l l operate at the f i f t h harmonic of the ion o r b i t i n g frequency (the RF power i s thus reduced by a f a c t o r of /5). The cyclotron performance may be improved by f l a t - t o p p i n g the RF voltage wave (Fig.4). The resonator design allows the addition of higher harmonics to the fundamental cavity mode. However, adding higher harmonics than h = 3 would require s p e c i a l treatment of the resonator surfaces and would severely increase the mechanical tolerances required i n order to a t t a i n high q u a l i t y factors f o r these higher harmonics. Also, the cyclotron operation i s not - 4 -much improved by e x c i t i n g resonator modes with h >y 5. For these reasons only the t h i r d harmonic of the fundamental, for which the RF cavity changes from a shorted A/4 l i n e to a 3/4A l i n e , w i l l be used. Adding about 11% of the t h i r d harmonic of the RF to the fundamental mode increases the microscopic duty f a c t o r and makes possible separated turn a c c e l e r a t i o n . The choice of the accelerating voltage i s determined by the av a i l a b l e RF power as w e l l as by the sparking c r i t e r i o n . Taking into consideration the power tubes a v a i l a b l e at the time of the TRIUMF proposal, the accelerating voltage was set at 200 kV. The tolerances on the RF voltage and frequency s t a b i l i t y are determined by the desired q u a l i t y of the outcoming beam. Both the resonator frequency and voltage amplitude w i l l be automatically c o n t r o l l e d . The frequency w i l l be held constant by means of tuning bellows mounted at the resonator root. Fast voltage amplitude control w i l l be achieved through screen modulation of the f i n a l tetrode stage (Fig. 5). A x i a l i n j e c t i o n of ions w i l l be used i n the TRIUMF cyclotron. According-l y , the resonator segments i n the c e n t r a l region of the cyclotron w i l l be modified to allow the i n s t a l l a t i o n of the centre post. A resonant transmission l i n e and a loop f o r coupling are being employed to transfer power from the main power am p l i f i e r to the Dees. The t h i r d harmonic mode w i l l be excited by means of a separate l i n e and coupling loop. - 5 -3. CRITICAL RF PARAMETERS It was necessary to prove that the resonator segments could be aligned mechanically so as to not a f f e c t the e l e c t r i c a l properties of the resonator system, i . e . mainly the q u a l i t y f a c t o r s , the voltage uniformity along the accelerating gap, the s t a b i l i t y of the resonant frequency, and the phase r e l a -t i o n between the upper and lower rows of the resonator segments and between the Dees. Another question was how good the contact between the hot arm t i p s and between the root pieces must be i n order not to lower the q u a l i t y factors and the voltage uniformity. Since no i n s u l a t o r s are being used, some mechanical o s c i l l a t i o n s can be expected. Should they produce a large frequen-cy v a r i a t i o n , dampers would have to be used to minimize t h e i r e f f e c t . We had to show that the frequency separation between the push-pull and push-push modes i s s u f f i c i e n t l y large so that no f l i p p i n g from one mode int o another could occur. It was also not obvious whether i t was possible to i n j e c t the two harmonics simultaneously into the resonator and yet achieve a steady state with a good phase and frequency s t a b i l i t y . The resonator t i p loading capacity as required by the condition of resonance i s not the same f o r both harmonics. For t h i s reason the t h i r d harmonic resonant frequency i s generally not three times as large as the f i r s t harmonic resonant frequency. A tuning element i n f l u e n c i n g one harmonic more than the other had to be found. The frequency tuning had to be i n v e s t i -gated f o r both the f i r s t and the t h i r d harmonic of the RF. The exact value of the RF fundamental frequency w i l l not be known u n t i l the main magnet tests are completed. The resonator must, therefore, be provided with some kind of tuning i n order to achieve isochronism. Two kinds of tuning were to be investigated - f i n e tuning and coarse tuning. Such a method had to be designed which would not cause the voltage v a r i a t i o n along the main gap to increase to a value greater than acceptable l i m i t s . - 6 -In the main cyclotron the beam load i s expected to be up to 1/4 of the t o t a l amount of the RF power. I f we allow such a beam of charged p a r t i c l e s to pass through a resonator, a l l possible harmonic modes of the fundamental cavity frequency can be excited, some of them undesirable f o r the RF system operation. The induced f i e l d gives r i s e to a periodic voltage V_ across the acce l e r a t i n g gap. Moreover, i f there i s an RF f i e l d i n the cavity during the passage of the beam, we also get a beam-RF f i e l d 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 or i n an absorption of the energy i n the e x i s t i n g RF f i e l d . In the case of the f i r s t harmonic of the RF, i t means that the beam absorbs a c e r t a i n amount of the RF energy. However, when both the f i r s t and the t h i r d harmonic of the RF are on, the beam may couple some amount of energy from the f i r s t into the t h i r d harmonic. Beside the two mentioned e f f e c t s , the beam can also cause a detuning of the resonator should the beam pulses not appear as a r e s i s t i v e load (zero phase angle). One other phenomenon - namely, multipactoring - i s associated with res-onator high voltage operation under vacuum. The multipactoring current due to o s c i l l a t i n g electrons increases very r a p i d l y because of el e c t r o n m u l t i p l i c a t i o n on the cavity walls. This current produces a heavy load on the RF generator, and i t i s often impossible to reach the operating voltage. Multipactoring can also lead to sparkovers because of ionized r e s i d u a l gas. The non-uniform shape of the resonators at the centre of the machine i s due to a cut-out which i s necessary to accommodate a centre post required for mechanical s t a b i l i t y of the magnet. Since the dimensions of the cut-out are comparable with the quarter wavelength of the t h i r d harmonic of the RF, the e l e c t r i c a l properties - the Q and voltage uniformity along the Dee gap - could be damaged by an improper shaping of the resonator segments. The o r i g i n a l proposal of the resonator system also c a l l e d f o r modified extreme end segments. Owing to a c i r c u l a r shape of the vacuum chamber, the segments were to be - 7 -tapered to f i t the vacuum chamber. This could again a f f e c t the t h i r d harmonic q u a l i t y f a c t o r . Some inve s t i g a t i o n s had to be made as to whether a resonant or non-resonant l i n e should be used to transfer the power to the Dees. By a proper choice of the parameters a very high standing wave r a t i o can e x i s t i n the resonant l i n e . With a high standing 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 varying loads, i . e . a sparkover or a reactive beam load giving r i s e to a mismatch i s easier to c o n t r o l . On the other hand, the main d i s -advantage i s i n the increased s i z e of the l i n e . The dimensions f o r the resonant l i n e must be chosen to prevent sparking and yet to tran s f e r the required power to the Dees. Computer programs had to be written to f i n d the l i n e parameters for a matched system, to ascertain the e f f e c t of a changed frequency on the transmission l i n e parameters, to f i n d how the system i s s e n s i t i v e to a varying beam load, and to investigate the resonances of the whole RF system. The question to answer was whether coupling to one Dee only i s s u f f i c i e n t to excite both Dees. A l l the mentioned problems had to be looked into before the f i n a l design of the resonator could be f i n i s h e d . The tolerances on the RF parameters, summarized i n Table I, are determined by the desired q u a l i t y of the beam,5 Table I I . - 8 -TABLE I Tolerances.on.the RF parameters Maximum duty factor Single turn extraction Frequency s t a b i l i t y Voltage s t a b i l i t y ( f i r s t ) Phase 3 r d to 1 s t (deg) Voltage s t a b i l i t y (third) Voltage asymmetry (cent r a l region) ± 1.25/10fc ± 2/10L ± 1 . 5 5/10* ± 7.5/10B ± 2.5/105 ± 0.15 ± 1/103 5/10 3 TABLE II Beam c h a r a c t e r i s t i c s F u l l width Spread Estimated Duty f a c t o r energy spread i n phase i n t e n s i t y (keV) (deg) (VA) (%) Raw beam ± 600 ± 12 200 7 ± 36 ( 3 r d ) 200 20 Low energy ± 220 ± 2 1 1.1 s l i t s 0.048 i n . 0.032 i n . ± 140 ± 1.8 0.5 1.0 ± 140 ± 14 ( 3 r d ) 16 8.0 Separated turn ± 80 ± 0.5 0.1 0.3 acceleration ± 45 ± 6 . 7 , ( 3 r d ) 1 3.7 ± 105 ± 6.7 4 3.7 F i n a l o r b i t ± 60 ± 0.5 0.05 0.3 s e l e c t i o n ± 5 . 0 , 0.5 2.8 ( 3 r d ) - 9 -CHAPTER I I . THEORETICAL CALCULATIONS  RESONATOR BASIC PARAMETERS AND RELATIONS 1.1 C h a r a c t e r i s t i c Impedance I f a low standing wave r a t i o e x i s t s i n the tr a n s m i s s i o n l i n e t r a n s f e r r i n g power to the Dees, an accurate value of (the c h a r a c t e r i s t i c impedance of the resonator represented as a trans m i s s i o n l i n e ) i s necessary f o r s e t t i n g the resonant l i n e parameters. A t i g h t c oupling between the top and bottom rows of e i t h e r Dee i s provided by means of f l u x guides (see F i g . 1). However, these f l u x guides, w i t h a 4 i n . gap, make c a l c u l a t i o n of more d i f f i c u l t . T h eir i n f l u e n c e on the c h a r a c t e r i s t i c impedance i s reduced as the number of resonator segments i n c r e a s e s . T r e a t i n g the resonator as a short ~ 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 , the c h a r a c t e r i s t i c impedance i s given by Z = A : + i^: a . D c • v G + jwC In a tr a n s m i s s i o n l i n e i n a normal use, G' fy 0, and i t f o l l o w s Z = Z - jZ f (1.2) c o J o 3 w i t h Y = a + j 3 , a = — , 3 = - , Z o = ^ . o R', L % G', C" are d i s t r i b u t e d parameters of the resonator a c t i n g as a transm 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 = i s the frequency. The value of a, the a t t e n u a t i o n constant (Appendix A ) , i s very s m a l l , and f o r t h i s reason we s h a l l assume Z = Z unless s p e c i f i e d otherwise. Since the c o capacitance per u n i t length may be w r i t t e n as C' = 8.854 e - x i o " 1 2  r g where w i s the width of the resonator, g i s the resonator gap, e i s the - 10 -d i e l e c t r i c constant of medium. The MKSA system of units 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 the resonator consisting of n segments i s expressed as where w = w .n , w i s the average width of resonator segment, w i s general-a a a l y d i f f e r e n t from the nominal width of a segment. 1.2 Tip Loading Capacity Although the approximate value of t h i s capacitance can be calculated by using several d i f f e r e n t methods, the exact value w i l l always be unknown unless we use a more precise c a l c u l a t i o n to obtain the correct f i e l d and p o t e n t i a l l i n e s . Since the resonator gap i s quite small, the ground arms w i l l have some e f f e c t on the hot arm t i p - t o - t i p capacity. The f i e l d l i n e s between two hot arm t i p s i n the r e a l resonator design would d i f f e r from the arrangement where the two hot arms are placed opposite each other i n a free space. Simultaneous-l y , other mechanical non-uniformities, such as beam probe housing, a f f e c t the true value. An over-estimate of the t i p - t o - t i p coupling capacity i s .made by assuming that the two hot arms are placed i n a free space. L i t e r a t u r e gives an expression r e s u l t i n g from the conformal mapping transformation which takes into account a transmission l i n e c o n s i s t i n g of two p a r a l l e l c i r c u l a r conductors. Replacing the two hot arm t i p s by two c i r c u l a r conductors of the same radius as that 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 two-conductor l i n e i s given by Z = 120TT K (1 .3) o w /D/2p + 1 / D/2p - 1 (1 .4) - 11 -where D i s the distance between the centres of the conductors and p i s the radius of conductor. Substituting D = 7.5 i n . and p = 0.75 i n . r e s u l t s i n Z = 275 ^ /m. The capacitance per unit length i s then C = 12.1 pF/m. For a fix e d hot arm width of 32 i n . , and considering both the upper and lower hot arm t i p s , we a r r i v e at = 19.68 pF. The measurements at hal f scale have resulted i n C^ ,_^  = 7.5 pF per resonant section, which means C^ ,_^ , = 15 pF at f u l l s c a l e . From now on we w i l l denote a t i p - t o — t i p capacitance between two resonant sections as 0^, . A resonant section includes both the upper and lower resonator segments. A hot arm tip-to-ground plane capacitance per one segment, which i s the same i n magnitude as w i l l be denoted as C^p. 1.3 Resonant Frequency For given values of the resonator c h a r a c t e r i s t i c impedance Z q , resonator length &, and the tip-to-ground loading capacity C^ -j-p' t n e frequency of the resonator o s c i l l a t i n g i n a push-pull mode i s determined by solving the follow-ing transcendental equation: I j Z tan\\ = 1 ( 1 ' 5 ) c I l^ QC T 1 T 1 This expression a c t u a l l y f i x e s a l l odd harmonic resonant frequencies, as w e l l . The push-push mode frequencies are calculated from the condition that -r = k'- ( 1 , 6 ) c 2 where k ' i s a p o s i t i v e integer and l* i s the distance between the root and the centre nodal plane. The equations above are v a l i d f o r a l o s s l e s s transmission l i n e . However, i f 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 high, at l e a s t of the order of magnitude of 1000, the resonant frequency of the resonator i s almost unaffected by the losses i n the d i e l e c t r i c . The frequency of damped o s c i l l a t i o n s i s given by - 12 -/ , 1 (1.7) Since the q u a l i t y f a c t o r i s of the order of magnitude of 7000, approximating the resonator frequency by aj Q i s j u s t i f i e d . From now on, co w i l l be used unless s p e c i f i e d otherwise. In the f i r s t approximation the resonant frequency depends on the t i p - t o -ground plane capacity as Afo _ "^TIP (1.8) CTIP 1 + — -s i n 2^ o-£ c S i m i l a r l y , we get Afo _ AZ 0 f o Z 0 1 + - * (1.9) s i n c where s i n 2 ^ £ = s i n 2d), <f> being the foreshortening angle. For the TRIUMF resonator the f r a c t i o n i n eqn. (1.9) i s le s s than 1/15 for the fundamental. 1.4 Resonator Power Loss There are two major energy losses i n the resonator which must be supplied by the external d r i v i n g mechanism. The f i r s t consists of the joule heating losses i n the resonator walls (skin l o s s e s ) . The second i s the power supplied to the beam and referred to the given beam current and energy. Variations i n the beam current r e s u l t i n corresponding v a r i a t i o n s i n the amount of power required to maintain the voltage amplitudes i n the cyclotron resonator at the s p e c i f i e d l e v e l . Moreover, a minor amount of power w i l l be provided i n order to compensate f o r the skin losses i n the resonant transmission l i n e and i n the coupling loop (see Section II.7.1). An expression f o r the resonator power loss due to the skin e f f e c t i s derived from the transmission l i n e equations - 13 -1 = V T sinh yx/Z Q + I cosh yx (1.10) V = Z Q I T sinh yx + V T cosh yx (1.11) which for a short circuited line simplify to 1 = 1^ cosh Y X V = Z I m sinh YX o T Since y = a + jg, a << 3,the resonator power loss is calculated from 1 2 ll^j cos2(-rx) dx o n |Vj sin 2-1 " 2^6w^ ( £ + — > ( 1 ' 1 2 ) a o c where y is the complex propagation constant, V^,I^, are the voltage and current at load (resonator root), |V | = Z ll^J is the voltage peak in the line which i s a quarter wavelength long, R' = is the surface resistance (fi/m) a for one panel, n is the number of segments,a i s the specific conductivity, i / S xl0" 2 6 = 6.62/ ^ O i s the skin depth and Z^ is the characteristic impedance of resonator segment. The power loss in the resonator oscillating in 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 in V = [V |-From the inspection of eqn. (1.12) i t follows that the skin losses in the third harmonic mode are higher by about a factor of / J than the losses in the fundamental mode, provided that a l l other parameters appearing in eqn. (1.12) are constant. Due to the large size of the full-scale Dees, each of them 3m x 16m, a few measurements had to be done at half-scale. For the tests the dimensions of the full-scale resonator segments were scaled down by a factor of 2 in order to obtain the same characteristic impedance as that considered for the main cyclotron. The power loss in the resonator modelled at half-- 14 -scale i s determined from pCl/2) = /2-pCD Now, should the two Dees be assembled and o s c i l l a t e i n a push-push mode the power loss i s determined from eqn. (1.12), where I = X/4 2 P = o,944 ^ i ! a L - (1.13) vTw Z z/oo a o cu where h = 1 f o r the f i r s t harmonic and h = 3 for the t h i r d harmonic. So f a r nothing has been said about the losses i n root pieces. The i n t e g r a t i o n i n eqn. (1.12), unfortunately, does not include t h i s small amount of power loss which i n the case of the TRIUMF resonator represents about 3% of the t o t a l RF power delivered to the Dees. This power loss i s calculated from 2nt / f i i 2 P = 3.776 Z C • | l | (1.14) w vaa a cu where t i s the height of a root piece, I i s the current through a root piece and o"cu i s the s p e c i f i c conductivity of copper. F i g . 6 shows the dependence of the resonator power on various parameters. The nominal values of parameters were: f =23.1 MHz, I = 3.0678 m, Z = 46 Q,, V . - V, = 100 kV, o o ol 1 C T Ip = 13 pF. V 1 i s the resonator t i p voltage (V = V i f (J) < 5 deg). 1.5 Energy Stored i n Resonator The voltage and current across the capacitance i n the resonant c i r c u i t are 90 deg out of time phase. This implies that the t o t a l energy i s a l t e r n a t e l y stored 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 stored i n a resonant c i r c u i t i s then found by determining e i t h e r the energy stored i n the c i r c u i t capacitance or inductance. The e l e c t r i c and magnetic energy d e n s i t i e s at any point i n the resonator are given by - 15 -= T C'|v(xj' w. H (1.15) (1.16) where V(x), I(x) are determined by eqns. (1.10) and (1.11) energy i s then calculated from The e l e c t r i c W = -E 2 C - | v(xj 2dx + \ C, V 2 TIP 1 1 Integrating and making use of eqn. (1.5), we obtain £ -s i n 2—£ c | v / + n s i n 2—1 c 4coZ„ (1.17) The magnetic energy i s calculated i n a s i m i l a r way W = ^ H 2 L'|l(x)| dx n 1 4 cZ, s i n 2—£ c (1.18) where we made use of Z l l j = |V | and Z = cL". Comparison of eqn. (1.17) with o T o o eqn. (1.18) reveals that the two expressions are i d e n t i c a l . It also follows from eqn. (1.18) that the s c a l i n g f a c t o r for going to a h a l f - s c a l e model i s equal to 1/2 because only the resonator length would be changed (except the frequency). S i m i l a r l y the energy stored i n the resonator o s c i l l a t i n g i n the t h i r d harmonic mode i s calculated from eqn. (1.18), and i s approximately equal to the energy stored i n the fundamental mode, provided that V o i s the same. The average energy stored i n the resonator for various values of resonator parameters can be evaluated from F i g . 6. 1.6 Resonator Quality Factor As the q u a l i t y f a c t o r increases the power which must be delivered to the Dees decreases. The high q u a l i t y f a c t o r also means that a l l resonant sections have been tuned to the same resonant frequency and that there i s the - 16 -uniform voltage d i s t r i b u t i o n along the main Dee gap. Two d i f f e r e n t q u a l i t y factors must not be confused, namely the q u a l i t y f a c t o r of the resonator i t s e l f , Q (unloaded q u a l i t y factor) and the q u a l i t y factor of the whole resonant system, Q^. The unloaded q u a l i t y f a c t o r of the resonator i s expressed by OJ W where W i s the energy stored i n the resonator i n resonance, P i s the power oo l o s t i n the resonator walls and f Q = — i s the resonant frequency. The s u b s t i t u t i o n for W and P (root losses excluded) leads to Z_,CT<$W O O a °-=—2~r— <1-19> where w i s the average width of a segment (taking into consideration the f l u x guides). The parameters i n eqn. (1.19) have been defined before. Since t h i s formula determines the q u a l i t y factor of the resonator o s c i l l a t i n g both i n the push-pull and push-push modes, we deduce that the d i f f e r e n c e i n Q's i s determined by the corresponding resonant frequencies of the two modes. In order to f i n d the s c a l i n g factor for a h a l f - s c a l e resonant cav i t y , i t i s s u f f i c i e n t to take into account the r e l a t i o n s derived i n previous paragraphs: Q ( l / 2 ) = 1 _ Q ( D The r a t i o between the fundamental and t h i r d harmonic q u a l i t y factors i s derived s i m i l a r l y : Q = /3 Q 3 1 A more precise c a l c u l a t i o n of the q u a l i t y factor i s done by use of the computer program RESLINE. The a d d i t i o n a l power loss i n the root shorting - 17 -plungers and i n the coupling loop i s then considered. To measure the q u a l i t y f a c t o r two d i f f e r e n t methods were used• Their d e s c r i p t i o n follows. The f i r s t method, the commonest i n microwave c i r c u i t s , i s based on measuring h a l f power points (or .707 voltage peak p o i n t s ) . Assuming that the resonator can be represented i n terms of lumped parameters R, L, C as a ser i e s resonant c i r c u i t , the power absorbed i n R i s given by the r e a l part of Iv I2 P = — (1.20) R + jRQ At resonance Ivl2 Re (P) ='^4 and at co = co^ and co = co^ ,C0O co J |vn|2 Re (P) = U and the Q i s determined from co Q = Q — (1.21) The second method makes use of the damped o s c i l l a t i o n s of the f i e l d s i n the resonator according to co / J -E(t) = E Q e" -2$- e 1 ^ 7 1 " ^ t (1.22) Taking the absolute value of eqn. (1.22), the Q i s given by Q = 2 ln(E /E) ( 1 ' 2 3 ) o The r e l a t i o n s h i p between the Q's and resonator parameters i s shown i n F i g . 7. - 18 -2. RF TOLERANCES 2.1 Frequency Detuning Due to Evacuation of the Vacuum Chamber During the evacuation of the vacuum tank the sides of t h i s tank tend to d e f l e c t inwards. Since the diameter of the tank i s much larger than i t s height, the d e f l e c t i o n of the tank l i d and bottom w i l l be mainly responsible for changes i n the alignment of the resonator segments. Suppose the resonator with a deflected arm can be approximated as shown i n Fig.8(b). Taking a constant t i p loading capacity, a new resonant frequency i s calculated from eqn. (2.1), which i s , i n f a c t , a transcendental equation 1 z ( n ) j U ) C T I P input (2.1) with z ( n ) = Z input o „(n-l) , . „(n) co t \ z - i + JZ ' tan -s (n) input J o c_ ;<n) + j z ( n - l ) t a n o input c Z a ) - 3 Z ( 1 ) t a n ^ s input J o c £ s = — n n >> 1 A small computer program, c a l l e d DETUNE, written to solve t h i s equation gave the following r e s u l t s - F i g . 9. If the d e f l e c t i o n +Ag corresponding to the evacuation of the vacuum tank i s known, one can then estimate the frequency s h i f t due to changes i n barometric pressure. I t i s f u l l y j u s t i f i e d to assume that the t i p loading capacity i s not influenced by changes i n pressure. A rough estimate has shown that a 1.27 cm d e f l e c t i o n of the hot arm t i p r e s u l t s i n an increase of the accelerating gap by 5.2 x 10 5 m. Then the resultant frequency s h i f t due to both the change i n the t i p loading capacity and the resonator length i s n e g l i g i b l e compared to that caused by the change i n - 19 -c h a r a c t e r i s t i c impedance. 2.2 Detuning of Resonator Due to Temperature V a r i a t i o n A change i n temperature causes the resonator panels to expand or to contract, r e s p e c t i v e l y . Since the root pieces are f i r m l y attached to the vacuum tank, the expansion (or contraction) of resonator segments can occur only i n the d i r e c t i o n of the accelerating gap. It i s supposed that the d e f l e c t i o n of the hot arm i s small and can therefore be neglected. The resonant frequency of the resonator o s c i l l a t i n g i n a push-pull mode i s given by Z tan ^1= — * — (2.2) ci c (si C. o ^ T I P where Z i s the c h a r a c t e r i s t i c impedance of resonator segment, f = i s the o o 2fr resonant frequency, % i s the hot arm length, C^j-p i s the t i p loading capacity per resonator segment, 2d i s the accelerating gap and w i s the resonator width. Assuming co = C O ( Z q ,C^^.p, £) , we obtain from the f i r s t d e r i v a t i v e of eqn. (2.2) the following expression (£ - c/4f Q) A Z Q A C T I P Al TT + „ + A f ^ Z Q C T I P I s i n 2 g » * fo i + * s i n 2-°-Ji c It i s easy to show that AC. TIP „ Aft Aw CTIP " d W ^ 2 . s _Aw Z D w Substituting into the above expression we a r r i v e at - 20 -Af A£ K where K = .1 s i n 2—% + Trd c_ = d_ £ s i n 2 ^ + TTJL A c So as a f i r s t order approximation we get f £ (2.3) This leads to the conclusion that the frequency s h i f t s caused by small changes i n the resonator width are compensated by the frequency s h i f t s due to simultaneous changes i n the t i p loading capacity. If the temperature of the resonator i s raised uniformly by an amount AT, the hot arm length w i l l increase r e l a t i v e l y by an amount aAT, a being the c o e f f i c i e n t of l i n e a r expansion. The expected frequency i s then determined from Af - aAT (2.4) For aluminum, a = 2.55 x 10 5/°C. The maximum frequency s h i f t for a given temperature change can be found i n Table I I I . TABLE III Percentage resonator frequency change vs temperature changes of resonator panels Af/f (%) AT (°C) -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 - 21 -C a l c u l a t i o n shows that eqn. (2.4) i s v a l i d for both the f i r s t and t h i r d harmonics of the RF. If the d e f l e c t i o n of the hot arm, as the temperature i s r a i s e d , cannot be neglected, the o v e r a l l frequency s h i f t i s given by a sum of frequency s h i f t s due to the hot arm expansion and due to the d e f l e c t i o n . For a small d e f l e c t i o n the change i n the hot arm length caused by a deflected arm i s small compared with the change i n length caused by an expansion. For a large d e f l e c -t i o n any change i n the length can be neglected because the frequency s h i f t caused by the d e f l e c t i o n i s at l e a s t an order of magnitude greater than that due to the change i n length. To estimate the frequency changecaused by the d e f l e c t i o n the parabolic approximation of the deflected arm i s considered. The r e s u l t s were obtained by means of the program DETUNE where the parabola was f i r s t approximated by a seri e s of s t r a i g h t l i n e s and then the same technique was used. The graph i n F i g . 9 summarizes the r e s u l t s . 2.3 Voltage V a r i a t i o n Due to Temperature V a r i a t i o n In addition to frequency detuning e f f e c t caused by resonator expansion temperature v a r i a t i o n s w i l l also cause changes i n resonator r e s i s t i v i t y . This w i l l produce changes i n resonator peak voltage. In the f i r s t approximation the r e s i s t i v i t y i s dependent on temperature as p = P q (1 + 3AT) (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 3 i s the temperature c o e f f i c i e n t of resistance. Should the temperature of the resonator r i s e uniformly by an 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 - p BAT o The s i m p l i f i e d expression f o r a resonator power loss i s given by - 22 -P = 2wa<5 where V Q - i s the resonator v o l t a g e peak, w^ i s the average width of resonator segment and <5 i s the s k i n depth. This equation can be r e w r i t t e n as P = P (1 + g AT) o (2.6) -3 5 Taking 3 = 4 x 10 f o r s i l v e r , P Q = 4.86 x 10 W f o r the main c y c l o t r o n (one Dee), V Q = 10 5 V and assuming a constant power input equal to P q i t i s p o s s i b l e to estimate the change i n resonator v o l t a g e caused by a temperature increase AT. from eqn. (2.7) V (2.7) V = / l + 3AT The r e s u l t s are presented i n Table IV. TABLE IV Resonator voltage v a r i a t i o n caused by temperature v a r i a t i o n AT (°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 2.4 Frequency Detuning Due to E l e c t r o s t a t i c and Magnetic Forces The resonator segment can be approximated by a number of 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 of charge r e s i d i n g on the surfaces of each c a p a c i t o r . 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 evaluated - 23 -as soon as one knows the charge d i s t r i b u t i o n i n the resonator. In the l i m i t when the number of capacitors goes to i n f i n i t y , one can replace the sum by an i n t e g r a l . The e l e c t r o s t a t i c energy i s then calculated according to eqn. (1.17). Since i t holds that Z = c C g where g i s the resonator gap and e i s the d i e l e c t r i c constant, and because the surfaces of the resonator are maintained at f i x e d p o t e n t i a l s by means of external sources of energy, the t o t a l f o r c e 9 acting between two resonator panels i s dW^  F = E dg 1 e ws * 7 s i n 2^1 1 + 2^ c ' o i (2.8) The average e l e c t r o s t a t i c force on the resonator panel i s 2TT/OJ F E V - 2 7 {o F E S ± ^ d t = \ F E (2.9) In the case of RF f l a t - t o p operation, the e l e c t r o s t a t i c force i s given by 1 e w a F E = " 4 V (1 + n) s i n 2—4, s i n 2 — I c c I + — ^ — + to, 2-±-OJ- V o i (2.10) where n =|^ 03/ v 0i|» ^ 0i» V Q3 a r e t ^ e fundamental and t h i r d voltage amplitudes i n the l i n e which i s a quarter wavelength long, 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 force acting on d i f f e r e n t unit areas of resonator surfaces i s not constant. I t depends on the distance from the root. Besides the e l e c t r o s t a t i c force, there i s also a magnetic force acting on the resonator panels. This force i s calculated i n a s i m i l a r fashion as the e l e c t r o s t a t i c force. The t o t a l magnetic force acting on the resonator - 24 -panel i s equal to the e l e c t r o s t a t i c force because of the equality of magnetic and e l e c t r i c energies at resonance. At resonance at any point i n s i d e the resonator the t o t a l ' energy density i s constant C ' | v(x)| 2+ L ' | l ( x ) | 2 = const. (2.11) where x i s the distance from the root. For t h i s reason the sum of the electro-s t a t i c and magnetic forces acting on a unit area of the resonator surface i s constant and does not depend on the distance from the root. The t o t a l force between two resonator panels i s F = F + F = 2F E H E The t o t a l average force i s F " = 1 F = F E = FH Evaluation of F for w = 32 i n . , g = 4.1 i n . , £ = 121.7 i n . and ct V , = V = 100 kV r e s u l t s i n F 3 V = - 5.46 Nt = - 1.23 l b . Since the t o t a l o l 1 average force per unit area i s constant the d e f l e c t i o n of the resonator panel c W due to t h i s force F can be calculated as i f the d e f l e c t i o n were due to the weight of the panel. The d e f l e c t i o n of the panel t i p i s given b y 1 0 Ag - K F a V (2.12) - 3 - 3 where K = 3 x 10 i n . / l b . I t follows then Ag = - 3.7 x 10 i n . = max - 3 -9.4 x io cm. The change i n the t o t a l average force due to t h i s displacement i s A F 3 V = 0.99 x 10~ 2 Nt = 2.23 x 10~ 3 l b . The d e f l e c t i o n of the resonator hot arm leads to a frequency s h i f t Af = - 0.01 % (see F i g . 9). Frequency changes were computed using the computer program DETUNE (see Section II.2.1). - 25 -2.6 Tolerances to be Met To maintain the resonator tuned f o r the desired operating conditions, the resonant frequency, voltage amplitude and voltage phase i n both Dees must be co n t r o l l e d . The sensors w i l l reference these quantities to some external reference. The RF system w i l l be kept i n tune by means of several servosystems. Their function i s to hold the proper phase r e l a t i o n between the Dee voltages, to hold a constant resonator frequency and to keep the f i n a l amplifier operating at the maximum e f f i c i e n c y at the resonator frequency. 1 1 During the RF f l a t - t o p operation i t w i l l also be necessary to control the phase r e l a t i o n and frequency r a t i o of both harmonics. The RF voltage w i l l be determined p r e c i s e l y by measuring the energy gain of p a r t i c l e s using beam probes. Capacitive pick-up probes w i l l also be i n s t a l l e d and they w i l l provide us with a c e r t a i n information on the l e v e l of the RF voltage. The a l t e r n a t i v e method of measuring the RF voltage would be to measure the Coulomb force exerted on a given surface S by an a l t e r n a t i n g f i e l d Vsinbot. This force i s given by F " 4 The d e f l e c t i o n would be converted into a s i g n a l to enable comparison with some reference point. However, the p o s s i b i l i t y of using both capacitive probes and probes measuring the d e f l e c t i o n due to the Coulomb force f o r a precise determination of the RF voltage i s hindered by two f a c t s . F i r s t , an accurate and permanent c a l i b r a t i o n i s d i f f i c u l t to achieve because of a possible motion of the ground arm and hot arm t i p s . The computer r e s u l t s i n d i c a t e that f o r a constant root current the t i p voltage changes by ±3% when a ±6 mm t i p d e f l e c t i o n occurs. A l i n e a r approximation of the deflected hot arm i s assumed. Second, the c a l i b r a t i o n i s influenced by the asymmetry i n the p o s i t i o n of a v i r t u a l grounding plane between the Dees. A frequency synthesizer w i l l be used f o r e x c i t a t i o n of the fundamental - 26 -and the t r i p l e r w i l l derive the t h i r d harmonic frequency. The fundamental resonant frequency w i l l be held constant by adjusting the p o s i t i o n of tuning bellows mounted at the root. The actual i n tune state w i l l be attained by holding a zero phase s h i f t between the root current and the loop current. The t h i r d harmonic voltage must be s h i f t e d by 180 deg with respect to the fundamental voltage. To achieve the a c c e l e r a t i o n of ions without any adverse e f f e c t s on t h e i r path, the voltage d i s t r i b u t i o n along the accelerating gap must be as uniform as p o s s i b l e . Simultaneously the asymmetry i n the voltages on the upper and lower resonator segments should be minimized and kept constant. Calculations 12 13 by Richardson ' which are summarized i n Tables V, VI give the allowable tolerances. le maximum TABLE V Permissible l e f t - r i g h t voltage asymmetry R (in.) i 15 25 35 50 60 Su (%) 0.5 0.5 0.9 1.9 2.7 where <5u = V, _ (R) - V . _ ( R ) l e f t r i g h t V n . ( R ) + V . , fR) l e f t r i g h t AV = V (R) - V, (R) upper lower The desired tolerances on the RF parameters are presented i n Table I (see Section 1.3). - 27 -TABLE VI Permissible top-bottom voltage asymmetry R (in.) 20 30 40 50 70 AV (%) 0.5 0.5 0.5 ' 1.0 2.5 3. VOLTAGE BREAKDOWN 3.1 Sparking Two major problems could be encountered i n achieving the resonator voltage l e v e l necessary f o r acceleration. These are sparking and multipactoring. Both sparking and multipactoring might lead to a voltage breakdown i n the resonator. We s h a l l f i r s t look at the p o s s i b i l i t y of sparking i n the resonator. According to K i l p a t r i c k ' s d e f i n i t i o n } ^ "current due to f i e l d emission i s considered necessary for sparking. In addition, energetic p a r t i c l e s are required to i n i t i a t e a cascade process which increases the f i e l d emission currents to the point of sparking". The l a t t e r i s also the reason why sparking i n the vacuum can occur at lower voltages than those which would be necessary i f only f i e l d emission i s considered responsible f o r i n i t i a t i o n of sparking. Sparking represents a spontaneous, abrupt, complete d i s s i p a t i o n of e l e c t r i c a l l y stored energy f o r a given voltage across the gap between two metal surfaces. For a cascade process to take place a c e r t a i n maximum energy must be supplied to one of the metal surfaces. At low gradients, of the order 10 V/cm and l e s s , ions present i n the gap supply the energy, but when high gradients e x i s t a cascade process i s due to p a r t i c l e s released from a metal surface by the thermal energy. - 28 -K i l p a t r i c k presents a c r i t e r i o n which determines a threshold below which no sparks should be observed i n vacuum (Presence of external magnetic f i e l d s i s excluded) : 1 . 7 X 1 0 5 W I = W E 2e E « 1.8 x 10lh (3.1) o W i s the maximum possible energy of a p a r t i c l e at the electrode surface p r i o r to a spark i n eV and E i s the f i e l d gradient i n V/cm. To f i n d the maximum RF energy W, a quantity V* representing the highest energy, n o n - r e l a t i v i s t i c multipactor voltage f o r TT deg t r a n s i t time i s given by V* = 2 2 2gTT| _ -1 N.R. (3.2) where g i s the gap i n cm, X i s the wavelength i n a free space i n cm, q i s the 2 charge of a p a r t i c l e and mQc i s the rest energy of a p a r t i c l e . I f the r a t i o V/V* < 1 f o r a given RF voltage peak across the gap, the energy W can be evaluated using the expression 2 W = 2 ZT7 m c 2 N.R. (3.3) Four d i f f e r e n t gaps are present i n the TRIUMF resonator and they are considered separately (see Table VII). It i s not known i n advance what kind of ions w i l l be present i n the resonator, however, one can consider protons which give the extreme upper energy W and the lowest factor V*. In other words, for heavier ions the s i t u a t i o n i s more favourable. Substituting the calculated values of E and W i n eqn. (3.1) re s u l t s i n values which are smaller than 1.8 x IO11*. In the case of a 1 i n . gap the energy W could not be evaluated because the r a t i o V/V* > 1. Another way of f i n d i n g W would have to be considered. However, i t i s believed that t h i s case i s roughly equivalent to the 2 i n . gap for which V/V* i s only s l i g h t l y l a r g e r than unity. For V/V* - 1 eqn. (3.3) gives only approximate values. The r e s u l t s are presented i n Table VII. Accordingly, no sparks are to be expected since the considered gradients are lower than those f o r which a spark w i l l occur as given by the K i l p a t r i c k ' s c r i t e r i o n . TABLE VII K i l p a t r i c k ' s c r i t e r i o n applied to TRIUMF resonator geometry g = 6 i n . g = 4.1 i n . g = 2 i n . g = 1 i n . V (kV) 2Q0 100 200 100 E (kV/cm) 13.1 9.6 39.4 39.4 X (cm) 1323.9 1323.9 1323.9 1323.9 2 m c o (MeV) 939.3 939.3 939.3 939.3 v* (kV) 1562.0 731.0 174.0 43.5 V/V* 0.13 0.14 1.15 2.30 W (keV) 1.02 0.54 * 9.16 -W I 4.1 x io 5 1.0 x 103 1.9 x 1011 o 3.2 Multipactoring Ranges To inv e s t i g a t e possible multipactoring regions i n the TRIUMF resonator 1 5 system a treatment s i m i l a r to that given by Smith w i l l be applied. M u l t i -pactoring (electron m u l t i p l i c a t i o n by secondary emission i n a vacuum) can s t a r t as soon as several conditions have been met. Many parameters such as the e l e c t r i c f i e l d , the resonant frequency, the resonator shape and others have a s i g n i f i c a n t r o l e i n defi n i n g the multipactoring region. If we have two metal surfaces i n a vacuum the time of f l i g h t of an elec t r o n t^ between the electrode surfaces must be an odd multiple of a h a l f period of the RF o s c i l l a t i o n - 30 -= (2n - 1) Tr 1 co f o r the given frequency, voltage and the gap for multipactoring to s t a r t . The second condition i s that the energy gained by the electron i n t r a n s i t must be s u f f i c i e n t l y high to give the secondary emission c o e f f i c i e n t greater than unity. For copper the energy range giving the secondary emission c o e f f i c i e n t greater than one i s 200 eV < W q < 1500 eV. The energy gain of the electron also depends on the i n i t i a l phase of the electron with respect to the RF voltage. Multipactoring usually places a heavy load on the generator and often prevents the build-up of the Dee voltage. The load presented to the generator may also be reactive i n character and therefore multipactoring may lead to detuning e f f e c t s . The lower and upper threshold multipactoring voltages f o r a copper material can be estimated using the following f o r m u l a e ^ , 0.0725 ( f B ) 2  T L 1 2 l l 2 {1 + [(2n - D - ^ r } . VTU < i - 8 5 VTL (3.6) where V , V T denote the upper and lower threshold multipactoring voltages. The multipactoring order i s found from l / 2 n S ^ V T L u 2V D (3-7) where V q , the energy picked up by the electron crossing the gap, i s greater than 100 eV for most materials. In the TRIUMF resonator system the following gaps are present: g = 6 i n . accelerating gap outside CR 4 i n . nominal resonator gap 3 i n . hot arm t i p to grounding plane (one Dee resonator) - 31 -2 i n . accelerating gap i n CR 1 i n . hot arm t i p to centre post (at i n j e c t i o n point) The f i r s t and the t h i r d RF harmonics have been considered separately. TABLE VIII; Threshold multipactoring voltages f o r the RF fundamental Gap (in.) VTL ( V ) VTU ( V ) Order 6 720 1340 1 4 320 590 1 3 180 330 1 2 80 150 -1 20 40 -TABLE IX Threshold multipactoring voltages f o r the t h i r d harmonic Gap (in.) V T L ( V> v T U (V) Order 6 6500 12100 1 2510 12100 2 1530 12100 3 1090 12100 4 4 2880 5360 1 1110 5360 2 3 1620 3000 1 620 3000 2 2 740 1360 1 1 180 330 1 - 32 -No higher order rtmltipactors can occur except those indicated above. A rate of r i s e of 1000 V/usec i s usually considered s u f f i c i e n t to pass through resonator multipactoring under good vacuum conditions. I f t h i s rate of r i s e i s a v a i l a b l e i t i s possible to break through the multipactoring range at poor -k - 5 pressures of the order 10 - 10 Torr. 3.3 Rate of Rise of Resonator Voltage The rate of r i s e of resonator voltage i s associated with problems of multipactoring. Knowing the transient build-up of resonator voltage amplitudes allows us to predict whether any d i f f i c u l t i e s w i l l be encountered during a switch on process. An i n v e s t i g a t i o n was c a r r i e d out i n order to compare the rate of voltage r i s e of resonator fed d i r e c t l y by a generator with that of a resonator fed v i a a resonant transmission l i n e . For t h i s an equivalence between a resonator with d i s t r i b u t e d parameters and a lumped parameter representation i s assumed. A resonator and a generator are coupled as shown i n F i g . 10(a), where R i s the i n t e r n a l resistance of the generator, V i s the generator voltage and to i s the resonant frequency of the resonator. The resonator i s represented by lumped constants as a s e r i e s resonant c i r c u i t . We do not wish to introduce any coupling network between the generator and the resonator as we are interested i n the rate of r i s e of resonator voltage amplitude without loading the resonator with a coupling network. For t h i s reason we w i l l use the E E equivalent c i r c u i t shown i n F i g . 10(d),where R , V_ are equivalent generator CJ CJ resistance and voltage obtained by transforming R , V v i a M. The e f f e c t of G G the coupling on the resonator frequency i s included i n a new capacitance C. The new resonator resonant frequency i s co = ° /Ec E The generator impedance R i s chosen so that the resonator i s matched for - 33 -only f = f Q . At t h i s point i t holds that R = R^. The condition that the steady state resonator amplitude at f = f be V r s 100 kV gives E V r V„ = 2 -v. S i n w t (3.8) where Q i s the q u a l i t y f a c t o r of the unloaded resonator and to = 2frf i s " the frequency of the generator. According to Kirchhoff's law for our c i r c u i t we can write AT i f E V r (3.9) L dt C v c Idt + R I + R I = 2 -r-^ s i n tot Assuming the system i s i n i t i a l l y quiescent,i.e. V„(0) = 0, 1(0) Laplace transforms r e s u l t s i n V„(s) 0 and taking P(s) where P(s) = (to2 + s 2 ) (LCs 2 + RCs + R^Cs + 1) V = 2 v c (3.10) The poles of V (s) are s, „ = ±ito 1.2 J 3,4 Having obtained the poles the inverse transform takes the following form 1 V0co e s t P(s) ds (3.11) This i n t e g r a l i s usually solved by c a l c u l a t i n g the value of the residue with respect to each pole and summing over a l l poles. I t follows then V. o , sk f c (3.12) k=l x-[P(s)] ds s=s, - 34 -To evaluate V~(t) a small computer program was written. The r e s u l t s are summarized i n F i g . 11, which shows the r i s e of resonator voltage amplitude during the transient f o r f = f = 22.66 MHz. The following lumped parameters determined from the power l o s s , energy stored, resonant frequency have been used for computation (CRM - one Dee resonator): C = 5.794 x 10 1 0 F L = 8.515 x 10~ 8 H, R = R^ = 1.709 x io"3 ft,, Q = 7091. The energy stored and power loss f or a steady state voltage amplitude of 100 kV have been computed by the program RESLINE. Since the q u a l i t y f a c t o r of a one Dee resonator i n the main cyclotron i s also the same, the rate of r i s e of voltage amplitude would also be given by the graph i n F i g . 11. To f i n d the transient response of the resonator voltage amplitude when the resonator i s fed v i a a resonant transmission l i n e the following equivalent representation of our system can be drawn - F i g . 10(b). V = V s i n ojt i s the Gr O generator voltage, R i s the i n t e r n a l resistance of the generator, co i s the generator frequency and U ) q i s the i n i t i a l resonant frequency of each c i r c u i t . Rj, C j , and R2, C 2, L 2 are lumped parameters of the resonator and the resonant l i n e , r e s p e c t i v e l y . It i s assumed that a steady state input resistance at f i s matched to the generator i n t e r n a l r esistance, i . e . R I 5 4f + R 2 = RG <3' 1 3> The mutual inductance, M = k / l ^ L ^ , i s determined from this condition. The c i r c u i t analysis y i e l d s the following two equations V i + c7 I 2 d t + ( L 2 + M ) ^ i - M | ^ + M — 2 - = 0 (3.14) f dlv d i d l 2 1 M ^ - M ^ + ( L 2 + M ) Z J L + _ I 2 d t + (R 2 + R G ) I 2 = V G (3.15) Supposing the system i s i n i t i a l l y quiescent the s u b s t i t u t i o n of Laplace transforms gives where - 35 -- V M C O C 2 " 2 C J | S 2  v ( S) = — ° - — (3.16) u l (co + s ) ( A 2 s + A 2s + A 3 s / + A^s + A 5 ) ( 1 - kz) A - 1 " i Q 2 + u 2 Q l R r A„ = — — + G 2 Q^gCl - k 2 ) L 2 ( l - k 2) w 2w 2 + Q!Q 2(w 2+ w 2) RgUj 3 0 . ^ ( 1 - k 2) Q j L g d - k 2) 3 1co 2(co 2Q 2 + 0) 1Q 1) R co2 A, = -2 + Y~ Q j Q i d - k ) L 2 ( l - k ) A. 2 2 co2 5 1 - k 2 In order to c a l c u l a t e the inverse transform of V (s) we must know the poles C l of V ( s ) . Since only two out of s i x e x i s t i n g poles could be evaluated C l numerically the computer was used to f i n d the remaining four poles. The inverse transform can be evaluated as 6 - V McoC0ojfco2s2 e S k t V, (t) = I o 2 1 2 k ( 3 > l y ) 1 k = 1 d 7 r p < 8 > W k where P(s) denotes the denominator of eqn. (3.16). The o v e r a l l r i s e of resonator voltage amplitude during the transient f or f = f^ = f 2 = f was found to be the same as that p l o t t e d i n F i g . 11. The i n i t i a l voltage r i s e i s plotted i n F i g . 12. The amplitude of the generator voltage, V , i s twice the amplitude at BB' (see F i g . 10(b)) f i x e d by the program RESLINE for a matched system. The parameters used i n computation have been f i x e d as follows. For a f i x e d i n t e r n a l resistance R = 2200 Q, (triode amplifier) a computer program RESLINE was run i n order to match the input impedance at f = 22.66 MHz. From - 36 -the values of energy stored, power diss i p a t e d , q u a l i t y factor computed by the program and using eqn. (3.13) the parameters M , C 2, k could be obtained: V G = 2 x 16422 V, Qa = 7091, Q 2 = 5473, M = 1.345 x 10 8 H, C 2 = 2.549 x 10~11+F, k = 1.048 x i o - 3 , L 2 = 1.935 x 10 _ 3 H . Similar analysis was done f o r a resonator c o n s i s t i n g of two Dees, where the representation i n terms of lumped parameters shown i n F i g . 10(c) was used. Choosing the current meshes as shown i n F i g . 10(c) r e s u l t s i n the following equations. 1 d l ! V l + L i d T - + c: r i d t + C " I 2 d t 0 (3.18) 1 1 1 1 1 f I,dt + — I 0 d t + I. dt + — I„dt + 1 2 c, • k-> 2 2 1 1 d l , d I 2 d t + I 3 d t + V 3 + < L 2 + M > d t ' I 3 d t = 0 d l , (3.19) (3.20) d l 3 d I 4 dl,. ! M _ _ M _ + ( L 3 + M ) _ _ + 3 • I.dt + (R 3 + RG)Ik = - V G (3.21) R^  , C j , and R 2, C^, are lumped parameters of the f i r s t and second Dee, res p e c t i v e l y . L 3 , C 3, R 3 are lumped constants of the resonant l i n e . C^ i s a Dee-to-Dee capacitance. M = kv/L~L„ i s the mutual inductance. It must be r 2 3 understood that the l i n e parameters L 3 , C g were calculated at the resonator resonant frequency f Q . The three resonant c i r c u i t s each having the same i n i t i a l resonant frequency were then connected together and driven at f Q . 1^(^ + 20^) L 2 ( C 2 + 2C^) L 3 C 3 o f i s not, i n general, the resonant frequency of three coupled c i r c u i t s . In thi s case, however, a proper r e s i s t i v e input impedance i s attained only at f (for a d e t a i l e d explanation, see Sections II.9.3 and II.10.2) - 37 -An i n i t i a l l y quiescent system is again assumed. A long and tedious algebraical derivation gave the following equations for V _ (t) and V _ (t) 8 R(s, ) e ^ V (t) = I — (3.22) k=l ^ ( B ) ] S where R(s) = - C 2 C 3 M V q s 3w(L l S + R2) (3.23) P(s) = (co2 + s 2 ) [ ( A 1 0 A 1 - A g A 7 ) s 6 + (A 1 1 A 1 + AJQA2 - \AQ)s5+ + (A 1 2 A 2 +-A11A2 + A 1 Q A 3 --AgA^s"* + + (A 1 2A 2 + A n A 3 + A I Q A 4 ) s 3 + (A 1 2 A 3 + A ^ + A ^ A ^ s 2 + ( A l A + A n A 5 ) s + A 1 2 A 5 ] (3.24) A = C L L (CC + C ) 1 2 1 2 H 1 2 A„ = (CTTC + C ) ( L R C + R L C ) 2 v H 1 2 1 2 2 1 2 2 A 3 = C H ( L 2 C 2 + L l V + R 1 R 2 C 2 ( C H C 1 + C 2 > + S ^2 ~ V Ak = C H(R 2C 2 + R 1C 1) + R x( C 2 - Cj) A 5 - C H " 1 A 6 = M 2C 2C 3 A 7 = L 1 ( C H C 1 + C V A = R (C C + C ) 8 1 H l 2 A 9 = C H A 1 0 = L 3 C 3 - 38 -A l l - C 3 ( R 3 + V A 1 2 = 1 s^, s^, ... , are poles of (s). The expression f o r (t) i s 8 R(s ) e 3 ^ V„ (t) = I — (3.25) k-1 i t P ( s ) ] k where R(s) = - MC 3V Qcos 2[(C t t + C 1 ) ( R 1 C 1 s + s 2 + 1) - C^] (3.26) P(s) = (u>2 + s 2 ) [ ( A 1 A 6 + A g A 1 2 ) s 6 + (A 2A 6 + A ^ y + A ^ A ^ + A ^ s 5 + (A_A„ + A A + A A + A A + A A + A A ) s 4 3 6 2 7 1 8 1 1 1 2 1 0 1 3 9 1 4 + (A,A 6 + A 3A y + A 2A 8 + A 2 1 A 1 3 + A ^ + A ^ s 3 + (A 5A 6 + AhA7 + A 3A 8 + A 2 1A 1 1 + + A ^ A ^ ) s 2 + (A 5A y + A^Ag + A n A 1 5 ) s + A 5A 8] (3.27) A, = C C,(L L - M2) 1 2 3 2 3 A 2 = C 2 C 3 ( R 2 L 3 + R G L 2 + L 2 R 3 ) A„ = L C + L C + C C R (R„ + R ) 3 3 3 2 2 2 3 2 V G 3 A, = (R_ + R )C + R C k G 3 3 2 2 A = 1 5 A 6 = C ^ C , + C 2) - 39 -C A.. = C (R_R + R R ) + L 1 14 3 G 2 2 3 2 \5 = R2 A = C 2C L 9 1 4 1 A, = C 2R C 10 1 1 4 A, = C C 11 1 4 A = C (L L - M2) 12 3 2 3 A, , = C„( R L + R L + L R ) 13 3 2 3 G 2 2 3 A computer program was written to evaluate the inverse transforms V (t) C l and V ( t ) . The r e s u l t s have indicated that V (t) and V (t) possess the S C l C2 same r i s e times, both o v e r a l l and i n i t i a l . The r e s u l t s are p l o t t e d i n Figs. 11 and 12. Note that replacing one Dee by a two Dee resonator at the end of the l i n e (CRM) lowers the input resistance by a factor of 2. In order to a t t a i n a resonator steady state amplitude of 100 kV, the a v a i l a b l e power from a matched voltage generator must now be doubled. Comparing the graphs i n F i g . 11 we deduce that the o v e r a l l r i s e of resonator voltage amplitude i s almost the same f o r both the resonator fed d i r e c t l y by a generator and the resonator fed v i a a resonant l i n e . This implies that the time constant of the resonator transmission l i n e system i s almost the same as that of the resonator fed d i r e c t l y by a generator. However, the i n i t i a l rate of r i s e of the resonator voltage amplitude i s less when the coupling between the r e s -onator and the generator i s achieved by means of a resonant l i n e (Fig. 12). This rate of voltage r i s e i s less than the desired value of 1000 V/usec. If the voltage source can not supply an a d d i t i o n a l amount of power i n excess of that needed f o r a steady state amplitude of 100 kV, the multipactoring range may not be passed. The lumped parameters were found by the program RESLINE (see Appendix A). - 40 -4. FREQUENCY TUNING 4.1 Root Shorting Plane Motion The resonant frequency of a resonator operating i n the push-pull mode vari e s according to eqn. (1.5) where Z q , C^-p, & a r e the possible variables which influence t h i s frequency. Tuning of the resonator by changing i t s phy s i c a l length i s s u i t a b l e both f o r f i n e and coarse frequency adjustments. Voltage uniformity along the accelerating gap i s unaffected by moving the root shorting plane. The q u a l i t y factors vary with a changing resonant frequency [see eqn. (1.19)]. Mechanical problems such as good contacts w i l l be the determining factors i n choosing t h i s kind of tuning for the main cyclotron resonator system. The graphs i n F i g . 13 present the r e s u l t s obtained by solving the above mentioned transcendental equation. The frequency was calculated as a function of length for f i x e d values of and Qp-j-p* 4.2 Capacitive E f f e c t Near the Accelerating Gap. The frequency tuning can be very simply accomplished by varying the t i p loading capacity at the high voltage end of the resonator. Solving the transcendental equation (1.5) with C^p as a v a r i a b l e gave the r e s u l t s plotted i n the graph i n F i g . 14. Several possible ways of introducing an a d d i t i o n a l capacitance e x i s t . In order not to a f f e c t the voltage uniformity t h i s a d d i t i o n a l capacitance must be spread uniformly along the accelerating gap. The resonant frequency of a ten section resonator made up i n h a l f - s c a l e was calculated when the t i p loading capacitance was increased by moving the cap a c i t i v e plates connected to the ground arm t i p s as shown i n F i g . 15(a). The following parameters were used i n c a l c u l a t i o n : p l a t e length = 4 m, plate width = .112 m, plate thickness = 0.012 m, resonator length = 1.52 m, Z q = 43.2 Q, Q = 5180, Q = 9020, C T I p = 7 pF. - 41 -The r e s u l t a n t frequency i s the same regardless of the number of sections considered i n the c a l c u l a t i o n . For s i m p l i c i t y one upper resonator segment i s assumed. An a d d i t i o n a l capacitance due to the plate i n the i n i t i a l p o s i t i o n d^ = 0.012 m i s C = 2.41 pF. By moving the capacitive plate down towards the hot arm t i p the t i p loading capacity i s increased by C = 0.413 0.0508 - d 0.0508 where d varies between d^ = 0.012 m and d 2 = 0.022 m. A computer program was used to solve the following equation Z tan —I = — - r r — - . ° C U TIP + C ) A new resonant frequency f r e s u l t s f o r each new value of C. The percentage frequency change i s defined as f " f l Af = 100 f l where f^ refe r s to the plate 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 function of the capacitive plate p o s i t i o n Ag = d - d j . 4.3 Ground Arm Tip De f l e c t i o n The i n v e s t i g a t i o n of tuning the resonator by changing the t i p loading capacity also included the d e f l e c t i o n of the ground arm t i p s . Two e f f e c t s are produced by d e f l e c t i n g the ground arm t i p s . Both the t i p loading capacity and the c h a r a c t e r i s t i c impedance of the resonator near the t i p w i l l change. These two e f f e c t s combine to give the new resonant frequency. In the f i r s t approximation the t i p loading capacity i s taken to be constant. For small d e f l e c t i o n s t h i s statement i s v a l i d . The new resonant frequency was computed by the program DETUNE. The deflected ground arm t i p was approximated by both - 42 -a s t r a i g h t l i n e and a parabola. Two sets of ca l c u l a t i o n s were done. The f i r s t one represented the coarse tuning of the CRM resonator. The resonator was represented by the following parameters: I = 3.09 m, AJ^ = 0.81 m, Z q = 38.3 ft, C T I p = 14 pF, f = 23.1 MHz f =69.27 MHz. AJL i s the length of the deflected t i p . The r e s u l t s are 0 3 1 B * p l o t t e d i n F i g . 17. The other set of calcu l a t i o n s simulated the tuning investigated on a h a l f - s c a l e resonator (see Section III.2.2). This c a l c u l a t i o n was done with the parameters: I = 1.55 m, Aft^ = 0.4064 m, C T Ip = 1 pF> Z q = 42.5 ft. In t h i s case the d e f l e c t i o n was accomplished by using hinged end sections of the ground arms as shown i n F i g . 15(b). The curves i n F i g . 18(a) represent the computed values. 4.4 Tuning Bellows at the Root It i s also possible to a l t e r the resonant frequency by changing the volume of the resonator. For a small change i n the resonator volume occurring 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 of a new resonant frequency. Suppose that a small portion of the root plunger (see F i g . 19) can be moved eit h e r forward or backwards. 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. A new resonant frequency i s given by f = f .: o < W H > ~ < W F > 1 + 2 — -<W> (4.1) where i s the magnetic energy i n the volume obtained by moving the bellows H inwards, 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 energy E stored i n the resonator. The following assumptions are made i n order to si 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 density i n the v i c i n i t y of the root plunger wE = 0. i i ) The magnetic energy density i n the volume V" i s the same as that at - 43 -the root. i i i ) The motion of the bellows from the i n i t i a l p o s i t i o n i s such that the resonator volume i s decreased, iv) The length of the resonator i s taken to be I - c / ( 4 f Q ) . It then follows that 1 f = f 1 + 4fD§ A i l + s i n 2^A£) c (4.2) where § characterizes the percentage portion of the root plunger surface which was moved and AJl i s the t r a v e l of the bellows. Given the values f =23.1 MHz, o 8 c = 3 x 10 m and § = 0.171 we a r r i v e at Af 1 + 0.052585 LI + s i n 2—Ail c 2% - 1 This expression y i e l d s the percentage frequency change as a function of the p o s i t i o n of bellows both f o r the f i r s t and t h i r d harmonics. The calculated values are presented i n Table X and p l o t t e d i n F i g . 20. TABLE X Percentage frequency change vs p o s i t i o n of tuning bellows AJl (m) A f 0 / f Q (%) - 0.002 0.0105 - 0.004 0.0210 - 0.006 0.0315 - 0.008 0.0421 - 0.010 0.0526 - 44 -4.5 Third Harmonic Tuning Diaphragms A section of the transmission l i n e , a quarter wavelength long, short c i r c u i t e d at one end and open c i r c u i t e d at the other one resonates at the fundamental frequency given by the length of the l i n e and at a l l odd harmonics of t h i s fundamental frequency. In the cyclotron two Dee resonator t h i s would represent the push-push mode which can not be used for acceleration of ions. For the f i x e d values of Z and C m T T 1 the -condition for o s c i l l a t i o n s of the o TIP resonator both at the fundamental and the t h i r d harmonic frequencies (push-pull modes) i s tan ^£ = 3 tan ^ (4.3) c c where co i s the fundamental frequency and £ i s the hot arm length. Solving this equation f o r an unknown hot arm length £ shows that no s o l u t i o n e x i s t s fo r the values of £ between 0 and -7- One concludes that the resonator can not 4 be excited both i n the f i r s t and t h i r d harmonic push-pull modes simultaneously unless one uses some a d d i t i o n a l tuning element. Two possible ways of i n f l u e n c i n g the resonant frequency e x i s t . A change i n e i t h e r a t o t a l capacitance or inductance r e s u l t s i n a d i f f e r e n t resonant frequency. Since the aim i s to retune either the f i r s t or the t h i r d harmonic for the required frequency r a t i o (4.4) a tuning element a f f e c t i n g one harmonic more than the other and placed at some distance from the root was sought. Movable diaphragms, connected to the ground arms, seemed to be the best s o l u t i o n (see F i g . 54). Two d i f f e r e n t methods were applied i n order to f i n d out the correct p o s i t i o n and estimate the detuning e f f e c t of diaphragms on e i t h e r harmonic resonant frequency separately. F i r s t perturbation theory was used to - 45 -c a l c u l a t e the frequency v a r i a t i o n f o r d i f f e r e n t positions and dimensions of diaphragms. The new resonant frequency i s determined according to eqn. (4.1) where now W' i s the magnetic energy i n a volume V'occupied by a diaphragm, n. i s 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 stored i n the resonator. For evaluation of energies stored i n a volume V the tuning diaphragm cross-section i s assumed to be rectangular. Since the f i e l d d i s t r i b u t i o n i s uniform along the v e r t i c a l a xis, the time average energies i n a volume V* are found by i n t e g r a t i n g eqns. (1.15) and (1.16) <w^> =|civol2c <w^> = | c i v / 5 u 2 - v + u 2 - V -s i n 2-f-l2 - s i n 2^o s i n 2 f k 2 - s i n 2 ^ (4.5) (4.6) where E, = g'/g, g i s the resonator gap, %' i s the height of the diaphragm, t = % 2 - £^ i s the thickness of the diaphragm and (£^ + 1^)12 i s the distance of the centre of the diaphragm from the root. The length of the resonator was taken to be £ = A /4 = 3A-/4, where A.., A are the wavelengths f o r the f i r s t , t h i r d harmonic, r e s p e c t i v e l y . The diaphragm width i s equal to the average width of the resonator segment. Substitution of the expressions given by eqns. (4.5), (4.6) and (1.18) into (4.1) leads to f = f 1 + 2? s i n 2-°-£ c 2 s i n 2-fLSLl 2^ ^c (4.7) F i g . 21 shows the v a r i a t i o n of f ^ , f 3 for a varying diaphragm p o s i t i o n i n s i d e the resonator. Af represents the change i n frequency caused by i n s e r t i o n of the diaphragm v e r t i c a l l y i n the resonator at a given distance from the root. From the r e s u l t s one would expect that the diaphragm placed e i t h e r at - 46 -A Q i/24 or 5X 0 1/24 from the root should make i t possible to simultaneously retune the c a v i t y frequencies so that the condition (4.4) i s met. However, t h i s could occur only i f the i n i t i a l frequency d i f f e r e n c e (measured) Af, - = |3f . - f I (4.8) 1-3 ' ol 0 3 ' i s smaller than Af otherwise the frequency change caused by turning the diaphragm would not s u f f i c e . Calculations were done with the given parameters: Z q = 38.3 Q, C T I p = 10.75 pF, g = 0.1016 m, %' = 0.06 m, t = 0.005 m, f = 22.66 MHz, f = 67.98 MHz. oi 0 3 16 Perturbation theory assumes a small portion of the o r i g i n a l resonator volume i s a l t e r e d . However, i f the height g' of the given diaphragm i s large, i . e . of the same order of magnitude as the resonator gap, a d i f f e r e n t method of perturbation theory must be used. A d d i t i o n a l e f f e c t s a r i s i n g from a capacity between the diaphragm and the hot arm might predominate. A resultant resonant frequency i s then determined by f i n d i n g new L, C parameters of the resonator. Supposing an equivalence between the resonant c i r c u i t s with lumped parameters and d i s t r i b u t e d parameters we represent the resonator as a lumped parameter resonant c i r c u i t . A t o t a l capacitance of an unperturbed resonator o s c i l l a t i n g at the push-pull frequency f = 22.66 MHz i s (see Section II.9.1) C - 1 2Z c o s i n 2-% to " 2 s i n - r % (4.9) where Z q i s the c h a r a c t e r i s t i c impedance , SL i s the resonator length and V i s the resonator peak voltage. Let the diaphragm represent a low frequency capacitance C H = 0.37 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 frequency capacitance increase ACy when moved to the v e r t i c a l p o s i t i o n at a given point i n the resonator (see F i g . 54). An o v e r a l l contribution to a t o t a l capacitance of the resonator i s given by - 47 -(AC y - C 2) s i n .-r^x s i n ^ 4 s i n .-j^ -x s i n > where x i s the p o s i t i o n of the diaphragm as measured from the root. C , are hot-to-ground arm capacities of a section where the diaphragm i s to be placed, i n the absence of the diaphragm. The values C u, (L , C„ can be calculated with n 1 2 a good accuracy. However, the value of AC^ should rather be measured as i t i s mainly a stray capacitance (the thickness of the diaphragm i s very small). For A f Q « f the new resonant frequencies are given by f„ = H 2 T T / L ( C + c^) 1 V 2rr/L(C + Cf. + O V rl (4.10) (4.11) Indices 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 of the diaphragm, res p e c t i v e l y . The resonant frequencies f o r the t h i r d harmonic are obtained using the same formulae with d i f f e r e n t parameters. The following parameters re used i n the c a l c u l a t i o n : Ac^ = 6 pF (measured), = 144 pF,C 3 = 143.9 pF, we £ = 3.19 m, g = 0.0635 m, t = 0.00635 m, w = 0.635 m. The graph i n F i g . 22 shows a percentage change i n resonant frequency when the tuning 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 p o s i t i o n . Since the resonant frequencies of the resonator are determined by the geometry of the resonator as w e l l as by the coupling network, two d i f f e r e n t cases must be considered. F i r s t the loaded resonator without a tuning diaphragm has such frequencies f ^ , f that an i n e q u a l i t y holds Af = 3f - f > 0 1 - 3 ol o3 (4.12) - 48 -It follows that i f a tuning diaphragm i s to be used, the f i r s t harmonic must be affected much more than the t h i r d harmonic i n order to s a t i s f y eqn. (4.4). Assuming Af ^ 0.3 MHz an element introducing approximately AC^ = 6 pF at i t s extreme p o s i t i o n should be used. I t seems that the best p o s i t i o n for placing a tuning diaphragm i n the resonator i s at 1.9 m from the root. This p o s i t i o n i s good for f = 22.66 MHz up to f = 22.66 MHz + 3% but not for ol. ol f Q = 22.66 MHz - 3%. In t h i s p o s i t i o n the RF fundamental voltage amplitude i s "only" about 78 kV compared with 96.5 kV at 2.75 m from the root (see Section III.2.6). If the resonant frequencies of the loaded resonator s a t i s f y an i n e q u a l i t y Af = 3f - f < 0 (4.13) 1 - 3 ol 0 3 one concludes that the t h i r d harmonic frequency retuning by means of a tuning diaphragm i s desired. The best p o s i t i o n i s now at about 0.5 - 0.6 m from the root. Taking the same |Af | $ 0.3 MHz the p o s i t i o n would be good for the range of frequencies f = 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 of the vacuum chamber demanded that the extreme end segments of ei t h e r Dee be tapered to f i t the vacuum tank. Results from preliminary measurements ( see Section III.4.1) indicated that the tapered segments influenced the e l e c t r i c a l properties of the whole resonator very s i g n i f i c a n t l y . The q u a l i t y factors were badly degraded. The reason f o r the drop i n the q u a l i t y factors seemed to be due to the transverse currents 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 design the tapered segment with dimensions such that the c h a r a c t e r i s t i c impedance would be constant throughout the segment. Suppose the dimensions of the extreme end segment hot arm are given as shown i n Fi g . 23(a) ( a l l dimensions are given i n cm). The required resonator gap i s found as follows. A sample c a l c u l a t i o n : 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 corresponding to this width i s In order to lower Z to obtain Z = Z = 49 0, i t i s necessary to choose the 1 1 o J resonator gap at th i s point i n the resonator to equal 49 gl = 5.2 = 2.85 cm The Table XI gives the required dimensions that 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. This change i n dimensions, however, increases the power loss i n an extreme end segment compared to the loss of a normal segment. The approximate c a l c u l a t i o n follows, where the parameters characterizing a - 50 -normal segment are taken to be: V - V = 100 kV, Z = 49 ft, a = 5.78 x 10 7 • o l o Mhos/m, $ = 9.8 x 10~ 6 m, f = 46 MHz, >t = 1.55 m, w = 0.403 m. According to o ' a . eqn. (1.12) the nominal power loss i n the resonator segment i s P = 14 kW. Calculated values f o r i n d i v i d u a l parts of the modified extreme end segment are presented i n Table XI, where an average width, w = b., was taken. The ° a I increase i n the power loss of such a segment was found to be AP = 6.5 kW, with the current density at the root going up by a factor of 2. TABLE XI Calculated gaps and power loss f o r a modified extreme end segment i b (cm) Z. (ft) l g± (cm) P ± (kW) 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 5.2 Central Region Segments The resonator segments i n the centre of the main cyclotron must be modified to allow the i n s t a l l a t i o n of the centre 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 . The basic c r i t e r i a which must be obeyed when modifying the c e n t r a l segments are: i ) A c e r t a i n minimum gap has to be preserved to avoid any p o s s i b i l i t y of sparking. i i ) The q u a l i t y factors should be affected as l i t t l e as possible, i i i ) The voltage v a r i a t i o n along the accelerating gap and around the - 51 -centre post must stay within the necessary tolerances (see Table I ) . The l a s t two requirements are aimed mainly at the RF t h i r d harmonic operation. Since the contour around the centre post i s fixed according to the requirements of beam dynamics, the possible modifications are l i m i t e d to the shaping of the t i p s of the resonator panels. This includes the change i n the actual t i p loading capacity and the change i n the resonator gap. A d e t a i l e d analysis was done to ascertain a l l e f f e c t s associated with shaping the resonator t i p s . The resonator i n h a l f - s c a l e i s assumed. i ) Let f = 3 f 1 and the top view of resonator be as shown i n F i g . 24(a). The following parameters are given: Z Q = 40 ft, f 1 = 45.32 MHz, &j = 1.56 m and = 1.433 m. To tune these parts of a-segment to the same resonance the required C^p per segment width i s given by eqn. (1.5) C T I P 2 = 7.98 pF C T I P 3 = 8.17 pF A C T I P = 0.19 pF *3 = 16 deg °TIP 1 = 18.8 pF C T I P 3 = 21.4 pF A C T I P = 2.6 pF * 3 = 37 deg <|>3 i s the foreshortening angle f o r the t h i r d harmonic. The d i f f e r e n c e i n the necessary t i p loading capacity for e i t h e r harmonic increases with an increasing foreshortening angle tb . Since tj>3 i s very large, small changes i n the t i p loading capacity produce large changes i n f 3 to s a t i s f y the condition of resonance. The requirement on t i p loading capacities can never be met simultaneously f o r both harmonics. One could f i n d some compromise in^the requirements f o r the t i p loading. Otherwise, the need for an a d d i t i o n a l tuning element a f f e c t i n g only one harmonic becomes more imperative i n order to obtain a proper frequency r a t i o , i i ) We want to f i n d the e f f e c t on the resonant frequency of e i t h e r increasing or decreasing the resonator gap near the arm t i p s . We assume that C^p - 52 -Z q , £ 2 are constant. The re s u l t s obtained with the computer code DETUNE show that an increase i n the resonator gap near the centre post r e s u l t s i n a p o s i t i v e Af and i n a negative Af .. The negative value of Af~ i s due to a large foreshortening angle CJJ^. I f the length of the resonator were close to ^  M the same d e f l e c t i o n would cause the frequency f^ to increase (see Section II.2.1). i i i ) A d e f l e c t i o n of e i t h e r the hot arm or ground arm t i p s influences the t i p loading capacity, as w e l l . An increase i n the resonator gap near the ti p s r e s u l t s i n lower tip-to-ground capacity. But a lower t i p capacity causes the frequency f^ to increase i n order to s a t i s f y the condition of resonance. As mentioned above small changes i n produce large frequency s h i f t s Afg. However, as stated i n i i ) an increase i n the resonator gap produces negative s h i f t s Afg. The resultant frequency w i l l be given by a combination of these two e f f e c t s . iv) By d e f l e c t i n g the resonator panels we not only s h i f t the frequency but we simultaneously change the voltage d i s t r i b u t i o n near the centre post. For given values of f , Z , l a p o s i t i v e d e f l e c t i o n (an increase i n the o o 2. resonator gap) gives a higher voltage at the hot arm t i p , while the negative d e f l e c t i o n leads to a lower voltage. I f the frequency changes i t i s not possible to predict the voltage d i s t r i b u t i o n unless one knows the f i n a l frequency. This d i s t r i b u t i o n also depends on whether the modified segment i s connected to the other segments or not. Both the frequency and voltage equipotentials can be influenced by the presence of other segments. We conclude that i t i s very d i f f i c u l t to pre-calculate the required t i p loading capacity when eit h e r the ground or hot arm t i p s are simultaneously deflected. The s o l u t i o n to th i s problem would probably .yield some optimum values of the Q's and voltage uniformity but t h i s would not be the i d e a l state - 53 -when a l l segments are tuned to resonance and a voltage d i s t r i b u t i o n i n the resonator follows a sine curve. That i s to say, we might obtain a uniform voltage along the accelerating gap but Q 3 could s t i l l be low. If we had a f u l l c o n t r o l of the t i p loading capacity when d e f l e c t i n g the panel t i p s the s i t u a t i o n would be more favourable. S t i l l the difference i n the t i p loading capacities AC^^ as mentioned i n i ) could cause other troubles. I t seems that the a b i l i t y to tune the segments to the same resonance by changing j u s t the t i p loading capacity i s better than i n the case when the hot arm and ground arm t i p s are deflected. Some approximate c a l c u l a t i o n s as to what capacitance i s required can then be done. During the tests (see Section III.4.2) the voltages at a given g r i d of points were measured [see F i g . 24(c)]. A computer program was written and used to p l o t the equipotentials. In the program the voltages i n each row are f i t t e d using a least square f i t . A p o s i t i o n i n each row i s then found which corresponds to a given voltage. A reasonable curve i s drawn through a set of points with the same p o t e n t i a l and t h i s curve i s i d e n t i f i e d with an equi-p o t e n t i a l . - 54 -6. BEAM LOADING If the i n j e c t e d p a r t i c l e s are not synchronized with the RF voltage phase i n the c e n t r a l region the t o t a l voltage on cavity i s both less than the applied voltage V and s h i f t e d i n phase with respect to i t . This i s due to the voltage Vg induced by the beam of charged p a r t i c l e s . I f we l e t the beam of charged p a r t i c l e s pass through an unexcited cavity the magnitude of V^ . i s very small. The amount of energy transferred through t h i s process w i l l be neglected. However, i f the c a v i t y i s excited by means of some external sources of RF energy the beam induced voltage V,, i s e s s e n t i a l l y determined through a resonator shunt resistance and a t o t a l c i r c u l a t i n g current i n the cyclotron. As a r e s u l t of t h i s beam-RF f i e l d i n t e r a c t i o n the beam can ei t h e r absorb or d e l i v e r a c e r t a i n amount of energy into the e x i s t i n g RF f i e l d s . 6.1 Beam-RF F i e l d Interaction A. The RF Fundamental Operation The Dee can be represented with lumped parameters R, L, C calculated from the known energy stored, power diss i p a t e d , q u a l i t y f a c t o r and the resonant frequency [see F i g . 25(a)]. Let the power input to the Dee be c o n t r o l l e d such that the amplitude i s always kept constant at a pre-determined value. Let the cyclotron d e l i v e r a beam I . Using the usual d e f i n i t i o n of current as:the rate of flow of charge across an i n t e r f a c e the c i r c u l a t i n g current i n the cyclotron becomes I c = N T I B (6.2) where N^ represents the number of times a p a r t i c l e s has orbited i n the magnetic f i e l d between i n j e c t i o n and ex t r a c t i o n . I t i s also assumed that there are no beam."losses i n the cyclotron. During the course of acceleration the p a r t i c l e s are not uniformly - 55 -d i s t r i b u t e d on each turn. They are concentrated i n bunches whose number i n each o r b i t i s found from a known harmonic acc e l e r a t i o n . We w i l l now derive an amplitude of current pulses (bunches) I which i n average give the current I A B [see F i g . 25(b)]. The condition of isochronism states that an ion completes one f u l l o r b i t i n T_ = ~z — = ~ (6.3) Ion f_ f. Ion 1 where f j Q n i s an ion r o t a t i o n frequency, f = h ' f - j . ^ i s a resonant frequency of resonator and h' i s a harmonic acceleration. A t o t a l number of p a r t i c l e s coming out of cyclotron per unit time i s Ng = 6.25 I x 1 0 1 8 p a r t i c l e s / s e c (6.4) A number of 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 T I o n ( 6 ' 5 ) and the p a r t i c l e s are concentrated i n h" bunches. We want to determine the amplitude 1^ (see F i g . 25) representing N^/h" p a r t i c l e s . Assuming that the p a r t i c l e s are uniformly d i s t r i b u t e d i n both areas we f i n d I* = I„ 21 (6.6) A B q + p If there are h/ bunches per cyclotron o r b i t then at any one time only l/h'* of the t o t a l c i r c u l a t i n g current i s undergoing acceleration during any h a l f cycle of the RF. During any complete RF cycle 2/h' of the t o t a l c i r c u l a t i n g current i s undergoing acceleration. Taking into account a complete course of acceleration the t o t a l current undergoing a c c e l e r a t i o n i s Ij- = 2 N T I B (6.7) where we c a l l I an i n t e r a c t i o n current. Consequently, the current amplitude - 56 -replacing a continuous current I^. i s defined as Since the cyclotron i s isochronous we can calculate for any i n j e c t i o n phase using N T = 4 e ^ s T <6"9> where Eg i s the f i n a l beam energy, V i s the resonator voltage and 8 = cot i s the phase angle between the maximum accelerating voltage and p a r t i c l e motion across the accelerating gap. We assume that the time taken to cross the gap i s vanishingly small compared to o r b i t c i r c u l a t i n g period. P a r t i c l e s which cross the accelerating gap at the RF voltage peak spend a minimum time i n the cyclotron before reaching the f i n a l energy. Those p a r t i c l e s coming ei t h e r e a r l i e r or l a t e r complete more turns i n order to reach the same f i n a l energy. We define an e f f e c t i v e voltage that accelerates a bunch of p a r t i c l e s as Iv l(sinp..+ sinq) V =|V I <cos cot> = (6.10) av 1 p + q v This allows us to c a l c u l a t e an average number of turns as N = ~ - (6.11) av 4e V av To be consistent with the f a c t that a bunch does contain p a r t i c l e s with 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) with N . In those J. c W cases where the phase width of a bunch i s very small use of would be j u s t i f i e d . Let us r e s t r i c t our attention to one RF cycle and assume the shape of a current pulse ( i . e . a bunch) as shown i n F i g . 25(b) and the RF voltage wave as shown i n F i g 25(b). In a complex notation V D(cot) takes the form R V R(cot) -|V2I ( 6 > 1 2 ) - 57 -Using Fourier analysis we can expand beam current pulses i n terms of sine and cosine functions [provided that f(u>t + 2TT) = £(u>t)] ao °° f (wt) = -r 1- V (a cosncot + b sinnut) (6.13) 2 n n n=l where a = A(p + q) O TT _ A(sin. np + s i n nq) n nu b = A(cos>'np - cos nq) n . nu The beam load per one RF cycle i s found by evaluating the i n t e g r a l f2Tr/to P" = V_(ait) f(u)t) dt (6.14) R 0 Integrating and multip l y i n g P' by f gives the complex beam power per unit time per one Dee P = P 2 + j P 2 (6.15) where P l = 2 r 7 ' V l ' ( s i n p + S ± n q ) (.6.16) P 2 = I^IV^cosp - cosq) (6.17) The expression (6.16) represents the amount of RF power delivered to the beam. Subs t i t u t i o n f o r A i n eqn. (6.16) would show that t h i s expression i s consistent with the usual d e f i n i t i o n of a beam r e s i s t i v e power given by I R E R P B = ^ T <6-18> However, we w i l l b enefit from our d e f i n i t i o n l a t e r . The imaginary component of P represents a reactive power stored e i t h e r i n a capacitance or an - 58 -inductance. A detuning of the resonator can be found as follows. Including the beam load i n our Dee representation y i e l d s the c i r c u i t shown i n F i g . 25(c) with l „ 12 2P (6.19) 1_J 2P (6.20) Since the resonator i s s t i l l forced to o s c i l l a t e at f = f , the phase angle of the t o t a l admittance Y I ' l l 1 R R_ j t C B J (6.21) i s calculated from tan $ = R X B ( R + V (6.22) A new resonant frequency i s determined from n 2CX. B (2CX B)' (6.23) B. The RF Flat-top Operation We w i l l now extend our analysis to the voltage waveform described by V„(cot) = IV Icosco-, t - |Vjcosto 0t K 1 -J- 3 3 In a complex notation t h i s becomes V R(cot) - | v i | e J u l t - | v 3|e j U3 t (6.24) A l l assumptions made i n part A. apply here. In addition we assume that the t h i r d harmonic resonant frequency f 3 s a t i s f i e s f =• 3f 3 1 (6.25) - 59 -and that the amplitude V 3 is maintained at a fixed level by means of control-led external sources. When calculating the amplitude A, eqn. (6.8), V must be modified as follows V =|vj <costo.t> - I v J <costo.t> av 1 l ' 3 3 - i , IV^Ksinq + sihp) - — (sin3q + sin3p) p + q (6.26) This value of V i s then used f o r c a l c u l a t i o n of N . A l l quantities and av av parameters have the same meaning as before. The complex beam power i s determined as i n part A. Performing the 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 f i n d I„E p = B B (sinp + sinq) + j(cosp - cosq) ^ ^ —I 3 F 2e £ (sinp + sinq) - -r-(sin3q + sin3p) p _ B^B _e (sin3p + sin3q) + j (cos3p - cos3q) T ~ 2e 3 e \o.za) (sinp + sinq) - — (sin3q + sin3p) with e =|V 3/ v j > 0. P^ and P^ are the complex beam powers for the RF fundamental and the t h i r d harmonic, r e s p e c t i v e l y . A negative value of a r e a l part of P^ means that the beam de l i v e r s a c e r t a i n amount of power into the t h i r d harmonic mode. By inspection of the d e f i n i t i o n of V we deduce that t h i s power i s taken from the f i r s t harmonic of the RF. However, the t o t a l amount of the RF power absorbed by the beam (both harmonics considered) stays constant per unit time. The beam load may thus be introduced by placing an a d d i t i o n a l negative resistance along with an a d d i t i o n a l reactance into the t h i r d harmonic c i r c u i t . The r e a l and imaginary parts of P and P are substituted into eqns. (6.19) and (6.20) i n order to c a l c u l a t e Rg, Rg, Xg, X^. C a l c u l a t i o n of Rg, Rg, Xfi, Xp enables us to f i n d a detuning of e i t h e r resonant c i r c u i t according to eqns. 3 - 60 -(6.22) and (6.23). A d i f f e r e n t lumped parameter representation must be used f o r the t h i r d harmonic. A subroutine BEAM was written and attached to the main RF program RESLINE. The graphs i n Figs. 26 - 31 correspond to a s i t u a t i o n i n which both harmonics of the RF are present. Operation with the RF fundamental only would y i e l d s i m i l a r r e s u l t s except that the power delivered to the beam from the RF fundamental would be constant f o r any p o s i t i o n of a beam pulse with respect to the RF voltage peak ( i n j e c t i o n phase). The voltage amplitudes are 113 kV and 13 kV for 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 i s set at 22.66 MHz. The pulse p o s i t i o n r e f e r s to the centre of a pulse. The phase width i s set to 80 deg. The following Table XII presents the approximate amounts of RF power required to cover the skin losses i n two Dees at a given peak voltage. To complete t h i s we note that the complex beam power P^ for the t h i r d harmonic voltage wave s h i f t e d with respect to the RF fundamental by a deg i s given by I E P = £ B B (cosa + j s i n a ) ( s i n 3 p + sin3q) - (sina - jcosa)(cos3p - cos3q) ^ 6 (sinp + sinq) - -j[sin(3q + a) + sin(3p - a)] (6.29) TABLE XII Resonator power loss for a given RF voltage amplitude Peak voltage (kV) Power loss (kW) CRM FIRST 113 145 CRM THIRD 13 4 MAIN FIRST 113 1250 MAIN THIRD 13 30 - 61 -6.2 Beam Induced Voltage I f we allow a bunched beam of charged p a r t i c l e s to pass through an unexcited resonator a l l possible harmonic modes of the fundamental resonator frequency can be excited. Some of them are undesirable for the RF system operation. The energy transferred to the resonator modes i s very small f o r the beam currents under consideration. For t h i s reason the p e r i o d i c voltages across the accelerating gap produced by induced f i e l d s can be neglected. However, i f the bunched beam passes through an excited resonator the beam-RF f i e l d i n t e r a c t i o n takes place (see Section II.6.1). As a r e s u l t the energy transfer occurs between the excited resonator mode and the component of the beam current which i s i n resonance with i t . The induced voltages appearing now across the gap are several orders of magnitude higher than those mentioned above. These voltages always oppose the p a r t i c l e motion. Their magnitude could have been determined i n Section II.6.1 from a known energy t r a n s f e r . Since we are also interested i n the transient build-up of these induced voltages a d i f f e r e n t approach has been chosen. Let us consider the RF fundamental operation. The Dee i s again represented with i t s lumped constants as a p a r a l l e l resonant c i r c u i t . The RF power input i s such that the steady state amplitude V = 100 kV i s attained. The power input i s kept constant. This assumption i s made only because we want to compare the magnitude of the voltage induced by the beam with V '. In Section II.6.3 i t w i l l be shown that instead of holding the power input constant the t o t a l voltage i n the fundamental mode must be held constant. I f the bunch of p a r t i c l e s a r r i v e s at the time of maximum RF voltage V„ = V . ,the r & R oi voltage Vg induced by the beam i n the fundamental frequency c i r c u i t i s 180 deg out of phase with respect to the fundamental voltage. Should the bunch not be synchronized with respect to the RF fundamental voltage peak the phase di f f e r e n c e between the induced voltage and the RF fundamental voltage would - 62 -l i e i n the range from 90 deg to 270 deg [see F i g . 25(d)]. This t o t a l voltage i s l e s s than the applied RF voltage V ^ and i s s h i f t e d i n phase with respect to i t . Let us turn our attention to F i g . 25(a). We can write I 1 - I 2 - I L + I C + I R ( 6 ' 3 0 ) where I , I are the RF external and beam current generators and I , I , I are 1 2 L L K branch currents. S u b s t i t u t i o n for current components leads to V 1 + 2aV 2 + w^Vj- £ (6.31) with ^ a = 2RC 1 = 1 , " I 2 I 2 i s defined by eqn. (6.13), 1^ i s given by I, = I cos wt (6.32) 1 o where to = t o ^ . We assume that the RF external generator I i s switched on at t = 0. Later when the steady state RF fundamental voltage has been reached the beam current generator with a current I 2 i s switched on at t = t , . Subst i t u t i o n of Laplace transforms i n eqn. (6.31) r e s u l t s i n V (s) = A + B + C A = -(6.33) C ( s 2 + U J 2 ) ( S 2 + 2as + to2) - t i s B = I b n n u ) s e 1 n=l C ( s 2 + n 2 t o 2 ) ( s 2 + 2as + t o 2 ) a . . 2 2 - t , s c _ _ £ a n n to e i  n=l C ( s 2 + n 2 t o 2 ) ( s 2 + 2as + t o 2 ) - 63 -V^Ct) i s found by evaluating the inverse Laplace transforms of eqn. (6.33). A i s responsible f o r a pe r i o d i c voltage on the ca v i t y which, i n f a c t , accelerates p a r t i c l e s . Only those components i n B, C, whose frequency coincides with u , i . e . n = 1, have an e f f e c t on the magnitude of the beam induced voltage V,,. The same procedure could be applied to the RF f l a t - t o p operation. The resultant equations f o r V 3 would be those given by eqns. (6.31) and (6.33) except f o r an opposite sign of I 3 and d i f f e r e n t values of a, R, C,L. The t o t a l voltage on the cavity i s then a sum of the t o t a l voltages i n e i t h e r harmonic mode, i . e . a sum of V „, V„ , V , V„ . ol B^ 0 3 B 3 A simple computer program was written to determine both the transient and the steady state response of i n d i v i d u a l components of a beam induced voltage. The r e s u l t s p l o t t e d i n graphs i n Figs. 32 - 36 show a transient t o t a l voltage response both i n the fundamental and the t h i r d harmonic modes when the beam current generator has been switched on. The graphs i n F i g . 32 correspond to the s i t u a t i o n i n which the RF fundamental only i s used f o r acc e l e r a t i o n of ions. Beam induced voltages during the RF f l a t - t o p operation are shown i n Figs. 33, 34. The steady state components of a beam induced voltage are presented i n the following graphs. F i g . 35(a) shows the applied fundamental RF voltage which can be compared with the voltage induced by the beam i n the f i r s t harmonic c i r c u i t shown i n F i g . 35(b). I f the beam arrives e i t h e r sooner or l a t e r , the induced voltage i s s h i f t e d i n phase and so i s a t o t a l voltage on ca v i t y . The steady state components of a beam induced voltage when the RF f l a t - t o p operation i s considered are presented i n F i g . 36. A d i f f e r e n t time scale i s used i n F i g . 36(b). A t o t a l voltage on cavity i s a sum of a voltage waveform due to external RF generators [see F i g . 35(a)] and beam induced voltages i n both the f i r s t and t h i r d harmonic c i r c u i t s . The parameters used i n the c a l c u l a t i o n can be found i n Appendices A, B, C. The steady state amplitudes due to external generators were V = 100 kV or V = 113 kV and - 64 -V q 3 = 13 kV f o r the fundamental or the RF f l a t - t o p operation, r e s p e c t i v e l y . 6.3 Concluding Remarks Due to the assumptions made (shape of a beam pulse, f i r s t to t h i r d harmonic frequency r a t i o , no beam losses i n the cyclotron etc.) the r e s u l t s represent an i d e a l s i t u a t i o n which w i l l hardly e x i s t i n the cyclotron. Since the beam pulse possesses a cer t a i n phase spread, the centre phase becomes very important i n centering the bunches with respect to the RF voltage peak. The centre phase can e a s i l y be found f o r any beam pulse which displays a symmetry with respect to some reference phase. However, even i f the beam pulse does not possess any symmetry i t i s always possible to f i n d a p o s i t i o n of the pulse with respect to the RF voltage peak which would r e s u l t i n a r e s i s t i v e beam, load (no frequency detuning). Throughout Section II.6.2 i t was assumed that a steady state value of i n t e r a c t i o n current was suddenly introduced into the cav i t y . For th i s reason the transient response of the t o t a l voltage i n e i t h e r harmonic mode shown i n Figs. 32 - 34 i s d i f f e r e n t from the r e a l s i t u a t i o n . However, the time i t takes to achieve a steady state value represents approximately the l e a s t time that would ever be needed. The assumption; i n Section II.6.2 that the power output from the external generator i s constant allowed us to compare the t o t a l voltages on the resonator with and without a beam. Unfortunately t h i s assumption would lead to the dependence of the t o t a l i n t e r a c t i o n current on the future values of the accelerating voltage. Instead, as i n Section II.6.1 we must assume that the t o t a l voltage i s always kept constant and i s always equal to the i n i t i a l RF voltage (without a beam). Let us consider the RF fundamental operation. The resonator i s matched to the power amplifier operating resistance by means of a resonant l i n e . The steady state RF voltage has been attained. The beam i s then i n j e c t e d e i t h e r at - 65 -a f u l l current or a f r a c t i o n of i t and the acceleration s t a r t s . If we assume that the i n j e c t e d beam i s centered with respect to the RF voltage peak, the beam load can be represented as an a d d i t i o n a l resistance i n our lumped parameter representation of a Dee. The t o t a l voltage on the resonator would tend to drop because of the beam induced voltage. The amount of the RF power deliv e r e d to the Dees must increase to compensate for the power extracted by the beam during the course of the a c c e l e r a t i o n . Thus, the t o t a l voltage on the resonator i s always kept constant. Moreover the parameters of the matching network, which i n our case consists of a resonant l i n e with three capacitors, must be readjusted because the shunt resistance of the resonator i s a l t e r e d . Computer r e s u l t s show that the phase angle of the impedance at the input of the transmission l i n e increases from zero to l e s s than two deg when the r e s i s t i v e beam load of 300 kW i s removed (or v i c e versa). However, the magnitude of the input impedance increases from 50ft to about 65 ft. For the CR cyclotron the e f f e c t of the beam load on the RF system i s n e g l i g i b l e . If the beam i s centered no detuning takes place. However, i f the beam i s not centered the a c c e l e r a t i o n can s t i l l take place despite the s h i f t e d resonator resonant frequency. As before, the power input to the resonator must increase. Since the resonator i s forced to o s c i l l a t e at the o r i g i n a l frequency the t o t a l voltage waveform i s not an exact sine wave. For small phase s h i f t s the r e s u l t s of our c a l c u l a t i o n are s t i l l v a l i d . Let us turn our attention to the RF f l a t - t o p operation. The resonator i s matched both to the operating resistance of the fundamental amplifier and to the operating resistance of the t h i r d harmonic amplifier by means of two separate transmission l i n e s . The beam i s now i n j e c t e d and centered with respect to the RF fundamental voltage peak. The beam induced voltage now has two components - one with the fundamental mode frequency and the other one with the t h i r d harmonic mode frequency. The fundamental component of a beam induced - 66 -voltage i s 180 deg out of phase with respect 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 voltage i s , however, i n phase with the o r i g i n a l t h i r d harmonic RF voltage. This implies that the t o t a l t h i r d harmonic voltage would tend to increase. The f i r s t harmonic transmission l i n e must be s l i g h t l y readjusted i n order to a r r i v e at a proper match. S i m i l a r l y the fundamental RF a m p l i f i e r must supply an a d d i t i o n a l amount of power which i s required i n order to accelerate a beam I to the f i n a l energy E . Jt> O But t h i s time the fundamental RF ampl i f i e r must also supply a second a d d i t i o n a l amount of power which i s absorbed by the beam. 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 of p a r t i c l e s , but i t i s delivered by the beam into the t h i r d harmonic mode. At high c i r c u l a t i n g currents the amount of power the beam de l i v e r s into the t h i r d harmonic mode becomes a s i g n i f i c a n t portion of the t o t a l RF power required to cover s k i n losses i n the t h i r d harmonic mode. The shunt resistance of 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 influenced by the presence of the r e s i s t i v e beam load. Retuning the transmission l i n e parameters i n order to match an alt e r e d resonator shunt resistance becomes more d i f f i c u l t . Besides the transmission l i n e retuning the power from the t h i r d harmonic amplifier must e i t h e r be reduced or some part of the t h i r d harmonic output power must damped i n a dummy load. This must be done i n order to maintain the t h i r d harmonic voltage peak constant. The amount of power which should be damped i s equal to the amount of power delivered by the beam into the t h i r d harmonic mode. The power delivered into the t h i r d harmonic mode i n the CR cyclotron i s very small and the computer r e s u l t s show that the input impedance i s only s l i g h t l y a f f e c t e d . If the i n j e c t e d beam i s not centered, a frequency detuning of e i t h e r harmonic mode occurs. Since the frequencies are fi x e d a phase s h i f t from the resonance occurs. The resultant voltage waveform i n the fundamental mode i s only 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 voltage waveform i n the t h i r d harmonic mode i s s e r i o u s l y d i s t o r t e d because the frequency detuning i s several times larger than that i n the f i r s t harmonic mode. I t i s also clear from Figs. 26 - 31 that while A f 1 i s p o s i t i v e , Af i s negative. This means that the resultant f l a t - t o p voltage wave with the presence 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 . This process leads to a worsening of the beam q u a l i t y and eventually to a complete loss of the beam. It could also happen that the t h i r d harmonic voltage i s i n i t i a l l y s h i f t e d with respect to the fundamental voltage wave. If the centered beam with respect to the fundamental voltage i s being accelerated the t h i r d harmonic resonant frequency i s changed. The resultant voltage f l a t - t o p wave i s again d i s t o r t e d by the t h i r d harmonic component of the beam induced voltage. In e i t h e r case, the subsequent behaviour of the cyclotron depends on the c a p a b i l i t y of the automatic c o n t r o l system. - 68 -7. COUPLING LOOP ASSEMBLY 7.1 Power Loss i n the Vacuum Seal Unlike the resonator system, which i s placed i n a vacuum chamber, the resonant transmission l i n e i s operated i n a i r . For t h i s reason a t r a n s i t i o n from vacuum to a i r has to be employed somewhere near the coupling loop. This vacuum feedthrough should serve two purposes: i ) give a vacuum-tight separation between the main vacuum tank and the transmission l i n e and i i ) help maintain the correct p o s i t i o n of the loop. The 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. The di s c i s stressed by mechanical loads (forces given from the Dee, by the shrinkage of the di s c at the assembly and the difference of pressure) and thermal stresses originated from a non-uniform temperature i n the body of the di s c and a restrained d i l a t i o n along the edges. The power loss i n d i e l e c t r i c material i s calculated from P (7.1) 1 2 2Rp where (7.2) Q 1 (7.3) tan 6 The capacitance between two c i r c u l a r conductors i s found as C = 2 i r £ r £ o t l n o y i y (7.4) The t o t a l power loss i n the d i s c i s determined from '2 1 (7.5) where t i s the thickness of the d i s c , R are outer and inner r a d i i , V i s - 69 -the voltage d i f f e r e n c e across the d i s c , i s the r e l a t i v e d i e l e c t r i c constant of the d i l e c t r i c material, e i s the r e l a t i v e d i e l e c t r i c constant i n the o vacuum, tan 5 i s the loss tangent., f i s the operating frequency and a i s the thermal conductivity. For the dis c made up from ceramic we have = 9 and tan 6 = 0.0002. Taking 2R2 = 6.5 i n . , 2R, = 1.5 i n . , f = 22.66 MHz, l v i = 9.1 kV (CRM), |V|= 12.7 kV (main c y c l o t r o n ) , t = 0.5 i n . and s u b s t i t u t i n g i n t o eqn. (7.5) r e s u l t s i n P = 5.1 tf. Using the d i s c with the same dimensions i n the main cyclotron would give P = 10 W. Taking t e f l o n material with tan 6 -0.0005 (most p e s s i m i s t i c value) and e = 2.1 we get for CRM P = 2.9 W. r & 1 However, i t should be understood that high power operation at a frequency d i f f e r e n t from the resonator resonant frequency might lead to a much higher RF loss i n the vacuum s e a l . I t i s shown i n Section II.10.3 that the r a t i o of voltage at the resonator t i p to voltage at loop, V../V , depends on the operating frequency. For lower values of t h i s r a t i o the operation with 100 kV on resonator t i p s can r e s u l t i n a voltage across the d i s c which i s increased by a factor of two over the computed value. I t i s then obvious that the amount of RF power loss increases. 7.2 Power Loss i n Adjacent Areas. The d i s c i s usually covered with s i l v e r on both the inner and outer edges to y i e l d a good and r e l i a b l e vacuum-tight s e a l . A c e r t a i n amount of power w i l l be l o s t i n these surfaces, which w i l l produce a d d i t i o n a l heating of the vacuum feedthrough. The power loss i n the c i r c u l a r conductor i s found from 1 t P c ; | 2 P 2 - I - 2 ^ ' 1 ' ( 7 - 6 ) where p i s a surface r e s i s t i v i t y , p = 2.6/f x 10 7 for s i l v e r , and I - I i s S S J-i the current through loop (input current). So we have for CRM P^ = 0.01 W i n the outer conductor and P = 0.05 W i n the inner conductor. Power losses i n - 70 -the transmission l i n e conductors above and below the ceramic d i s c are calculated according eqn. (7.6), where we take t = 23.5 cm. A s u b s t i t u t i o n y i e l d s P 3 .= 0.17 W for the outer conductor and P 3 = 0.44 W for the inner conductor. Since the current through the loop i n the main cyclotron i s ten times l a r g e r , the values of P^ and P^ would be increased by a f a c t o r of approximately 100 provided that the same dimensions are used f o r the main cyclotron. Once again the values of P^ and P^ are s i g n i f i c a n t l y influenced i f the operating frequency i s d i f f e r e n t from the resonator frequency. 7.3 Power Loss i n the Coupling Loop The power loss i n the coupling loop i t s e l f i s produced by the input current and by the c i r c u l a t i n g current i n the resonator. The input current i s , i n f a c t , responsible f o r the d e l i v e r y of the required amount of power to the Dees. This input current i s spread over most of the lower part of the coupling loop (that part of the loop facing the ground arm). This f a c t must be taken in t o account when computing the power loss due to the input current. Let %' be the length and w' be the width of the coupling loop. The power loss due to the input current i s then calculated from P, = ^ R U J 2 (7.7) k z s L % " where R = p —- i s the surface resistance and I T i s the loop current. We s s w L also know that at a given point i n the resonator the magnetic f i e l d i s constant i n the v e r t i c a l d i r e c t i o n . I f the coupling loop i s inserted the magnetic f i e l d l i n e s are d i s t o r t e d . As a f i r s t approximation we assume that the f i e l d density at the surface of the coupling loop i s the same as that at the c a v i t y w a l l . This implies that there i s also the same current density. The power loss due to the resonator c i r c u l a t i n g current i s found as (7.8) where I i s the current i n a s t r i p of the resonator w a l l . The width of the w s t r i p i s equal to the loop width. Both P^ and P 5 are calculated by the code RESLINE. The operation of the system at a frequency d i f f e r e n t from the resonator resonant frequency would a f f e c t the value of P^. Since the reactive component of the resonator shunt ( p a r a l l e l ) impedance becomes comparable with the r e s i s t i v e component, the magnitude of the impedance presented to the l i n e at a coupling point i s decreased. However, the resonator voltage step-up r a t i o k i s p r a c t i c a l l y constant i n the v i c i n i t y of the resonator frequency. This means that the magnitude of I i s increased which increases the amount of power dissipated i n the coupling loop. This increase i n P^ may reach a few Watts i n the CR cyclotron, but a few hundred Watts i n the main cyclotron. - 72 -8. TRANSMISSION LINE The main factors which influence a choice of the resonant l i n e parameters are the maximum power that can be transmitted and, consequently, the maximum permissible voltage. Since the l i n e i s operated i n a i r the p r o b a b i l i t y f o r getting sparks due to moisture, sharp edges etc. i s much higher. The maximum voltage i n the l i n e depends mainly on the load connected at the end of l i n e . The maximum voltage gradient happens to be near the inner conductor and i s given by E = Ra l n ( R 2 / R l ) (S- 1) where V i s the voltage d i f f e r e n c e between two conductors. Basic parameters s i m i l a r to those derived i n Section II.1 could be found, however, since the l i n e i s terminated i n a general complex impedance the expressions would become quite complicated. Therefore the q u a l i t y f a c t o r , power l o s s , energy stored and other parameters are calculated accurately using the computer programs RESLINE and MATCH. B a s i c a l l y the power loss and the energy stored are calculated by int e g r a t i n g the products R III 2 , L l l l 2 , C l v l 2 u s i n g (1.10) and (1.11). The method f or computing the lumped constants of the resonant l i n e i s outlined i n Section II.9.3. Any simple c a l c u l a t i o n of the l i n e ' s properties i s also hindered by the presence of three capacitors which s i g n i f i c a n t l y a f f e c t the e l e c t r i c a l properties of the l i n e . The capacitor near the loop helps adjust the loading impedance as desired; the capacitors i n the centre of the l i n e and at the tube tune the l i n e to a t t a i n a desired r e s i s t i v e input impedance ( i . e . to achieve a resonance of the whole system). In addition, the centre capacitor shortens the l i n e . The e l e c t r i c a l l i n e length (capacitors and loop included) i s nA/2. For the given values of c h a r a c t e r i s t i c impedance, length and loading impedance there always ex i s t s a combination of the three capacitors such that the - 73 -required input impedance i s obtained. A resonant l i n e i s desired when dealing with an unstable load because i t reduces the e f f e c t , at the tube, of an altered l i n e terminating load. Consequently, the tube parameters may have to be readjusted only s l i g h t l y . In addition, sparkovers i n the resonators do not present the problems which would e x i s t with a non-resonant l i n e . Such arc-overs cause a detuning of the resonator and, therefore, the system i s immediately mismatched, i . e . a match between the tube operating resistance and the resonator shunt resistance no longer e x i s t s . Hence, under these conditions, only a f r a c t i o n of the resonator energy may reach the power tube. As i t i s anticipated that the resonator q u a l i t y factor 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 value an i n v e s t i g a t i o n was c a r r i e d out to estimate 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 taking into account a t h e o r e t i c a l value of the Q. The parameters were then fixed and the input impedance was computed f or d i f f e r i n g values of the resonator q u a l i t y factor. The r e s u l t s show that the input impedance varied f o r the values of Q from 7000 down to 4000. However, the phase angle of the input impedance was always close to zero. The conclusion i s that the parameters of the l i n e must be readjusted i f the power am p l i f i e r i s to operate into a given resistance. I t was also noted that the voltage at the input of the l i n e was almost unaffected by the change i n Q. The computer program RESLINE was also applied when the r e l a t i o n between the resonator peak voltage and the tube parameters was investigated. The system was f i r s t matched for the resonator fundamental voltage amplitude of 100 kV and the l i n e parameters were then f i x e d . I t was found that f o r up to a 10% change there was a l i n e a r r e l a t i o n s h i p between the resonator fundamental voltage-, peak and the voltage and current at the transmission l i n e input. The change i n phase angle of the input impedance was n e g l i g i b l e . - 74 -The r e s u l t s of the computer program RESLINE r e f e r to the operation of the system at the resonator resonant frequency f . Should a proper match ( i . e . a r e s i s t i v e input impedance) be attained at f =f f the power loss i n the l i n e may be e i t h e r higher or lower than f o r the case f = f . Computer r e s u l t s show that i f the i n i t i a l match i s obtained at f = 22.652 MHz < f = 22.660 MHz the o voltage across CP(2) i s 14.6 kV when the resonator t i p voltage i s 100 kV (CRM). Should we match the l i n e at f = 22.668 MHz > f = 22.660 MHz the voltage across CP(2) would be 2.5 kV. The voltage i n the f i r s t section increases with an increasing Z . The capacitor voltage at f = f i s 8.4 kV. In the main o o cyclotron the operation at f = 23.096 MHz < f = 23.100 MHz would r e s u l t i n the voltage across CP(2) of about 34 kV while the nominal value at f = f i s o 13.1 kV. - 75 -LUMPED PARAMETER REPRESENTATION 9.1 Resonator Lumped Constants To enable us to ca l c u l a t e such quantities as the rate of r i s e of resonator voltage, beam loading, resonator voltage and input impedance as functions of generator frequency the resonator had to be represented i n terms of lumped parameters R, L, C. Their d e f i n i t i o n follows immediately once the magnetic and e l e c t r i c energies stored i n the resonator have been determined. Let the resonator voltage peak be V,. The resonant frequency of a resonator made up of n segments i s f Q . The lumped capacitance i s r e l a t e d to the resonator peak voltage V by wE = i c l v i | 2 (9.1) so that c - f c s i n £ + s i n c - 2 (9.2) where C' i s a capacitance per unit length of a resonator segment and V, = V sin—^ S L . The lumped inductance i s then calculated from 1 o c 1 L = wo2C (9.3) S i m i l a r l y the lumped inductance i s rel a t e d to the resonator peak current I by (9.4) This r e s u l t s i n 2 n £ + s i n 2 — V c_ 2 ^ c (9.5) Consequently, the lumped capacitance i s given by 1 C = (9.6) % L - 76 -A l l our ca l c u l a t i o n s use lumped constants L, C as defined by eqns. (9.2) and (9.3). L, C parameters are unaffected by a choice of representation. However, a s e r i e s resistance must be used when the resonator i s represented with a ser i e s resonant c i r c u i t and a p a r a l l e l resistance i s to be used when replacing the resonator with a p a r a l l e l resonant c i r c u i t . The p a r a l l e l resistance i s defined by h-^-u (9-7) which y i e l d s a f t e r s u b s t i t u t i o n 2o« w Z 2 s i n 2 ^ % R - - 3 ° C (9.8) 2_e. c The s e r i e s resistance R i s defined i n a s i m i l a r way by A l l lumped constants are computed by the program RESLINE. 9.2 Representation of the Coupling The RF energy must be supplied to the Dees by means of some coupling network. The most common coupling i s made by means of e i t h e r a capacitor or a coupling loop. We s h a l l r e s t r i c t ourselves to inductive coupling by means of a coupling loop. Let us consider loop coupling to the resonant cavity as shown i n F i g . 38(a). The resonator i s represented by a serie s resonant c i r c u i t with lumped constants R, L, C re l a t e d to the resonator peak voltage V (V^ = V^sin cot). L^ and R^ are the loop self-inductance and the loop resistance, r e s p e c t i v e l y . I ' i s the current i n the resonant c i r c u i t , V i s K L the voltage induced i n the loop (jcoL I i s not included). M = k V L L i s the Li Li L> - 77 -mutual inductance, k/ i s the coupling c o e f f i c i e n t , k i s the resonator step-up r a t i o . The impedance Z " seen at the input terminals BB" i s where the t h i r d term i s i d e n t i c a l with Z' given by eqn. (10.18). This impedance must now be matched by some means to the power tube stage impedance. Two methods may be employed for t h i s purpose. F i r s t when a non-resonant l i n e i s used, the value of the mutual inductance must be chosen so that the r e s i s t i v e part of Z"' i s equal to the c h a r a c t e r i s t i c impedance of the l i n e . At the same time a capacitive component has to be introduced at the loop i n order to cancel the inductive reactance of the loop. The other method, described i n Section II.10.1, makes use of a resonant l i n e with parameters chosen so that the l i n e merely transforms the impedance Z into the tube impedance. At the resonant frequency, f , the resonator presents a pure r e s i s t i v e load at the coupling point. The system which consists of the resonator with a coupling loop and a resonant l i n e must be tuned to the resonant frequency at which the resonator o s c i l l a t e s i f l e f t unloaded. As shown i n Section 11.10.2 the whole system i s then resonant at the resonator frequency f . Only then can a power transfer and a resonator voltage be achieved with transmission l i n e parameters as computed by the program RESLINE. At resonance f = f and we have (9.10) o L (9.11) and M d l dt (9.12) - 78 -so that M = L r k (9.13) The same definition of M follows from the equations of a shunt (parallel) resonator resistance. At resonance the shunt resistance seen at a coupling point i s 2P (9.14) where P is the resonator power loss. From eqn. (9.10) R^  must also be equal to 2 2 M to o R (9.15) This leads to M = L R This definition of M is consistent with calculation of the voltage V induced Li in the coupling loop. The coefficient k' can be expressed as R (9.16) In the program RESLINE, the mutual inductance i s computed according to v r - 3 b M = y 0 nw ° (9.17) where a, b are loop dimensions, w is the average width of the segment, n is a 7 the number of segments, U Q = 4TT X 10 H/m. One must bear in mind that a l l computing is done in terms of distributed parameters. For this reason the current through the root is given approximately by (9.18) I J = I = where I Q , V q are voltage and current peaks in the line which is a quarter wavelength long and V = V Q s i n ^ i . Assuming now that the resonator has been - 79 -replaced with i t s lumped parameters R, L, C where L = i o D 2 C we f i n d that the current i n the resonant c i r c u i t i s (9.19) This i s d i f f e r e n t from I The two expressions for a mutual inductance are r e l a t e d to one another by the following formula I M = M av R (9.20) where I i s the average current flowing i n the c a v i t y along the coupling loop av (the current which i s taken for computing the loop induced voltage V ) Let us return again to a diagram shown i n F i g . 38(a). Applying Kirchhoff's equations to t h i s system we f i n d the current through the loop i s where I = V L L R Z DZ T + w2M2 (9.21) Z R = R + 3 u)L - Ceo Z L = *L + J W L L which for f = f reduces to o I, = V. R L L R R + to2M2 + ico L R L o o L (9.22) Consequently, the phase s h i f t I and V' i s Lt Li tan $ = co L T o L 1 R L + B P (9.23) S i m i l a r l y we obtain I R - 80 jioM R L ZZ + 032M2 L K 03 = 03Q = - V (9.24) L ^ R + 032M7 + j 03_L R L o J o L A phase s h i f t between VT" and I " at resonance i s t a n $ 2 = _ _ ^ (9.25) ° L Since the resonator voltage i s given by V v = _ _k_ l^M R jcoC Z„Z T + 03ZM2  J R L 03 = 0)Q V L M 1 2 M 2 4. T „ (9-26) C R LR + O J 2 M Z + J0) oL LR the phase angle between V and I i s found to be R Li 1 03L -t a n ^ =  = 0 (9.27) 03 = 03„ The phase angle between I ' and V i s found as being i d e n t i c a l l y equal to 90 deg tan <& -> «> (9.28) If the Q i s low the resonant frequency given by eqn. (1.7) should be considered. As a r e s u l t a l l vectors would be s h i f t e d s l i g h t l y . 9.3 Lumped Constant Representation of a Resonant Line There are several possible ways of representing a resonant l i n e by lumped parameters at i t s resonant frequency, f . In order to represent the l i n e i n a wide frequency range by lumped parameters, a number of lumped constant c i r c u i t s would have to be used. Since we are mainly interested i n the behaviour of the l i n e i n the v i c i n i t y of f the representation of the resonant l i n e by a s i n g l e lumped constant resonant c i r c u i t i s adequate. - 81 -Let us consider the following representation [Fig. 38(b)] where R^, C , L^ are known lumped constants of the resonator consisting of one Dee, R^, C^, L^ are unknown lumped constants of the l i n e , M " i s the mutual inductance. The parameters M", R^, C^, L^ are f i x e d as follows. Suppose we know the average e l e c t r i c energy stored i n the l i n e <W'>, the average magnetic energy stored i n the l i n e <W">, tube voltage and current V_ T T T )„, I_ 7 r D„, and the q u a l i t y f a c t o r rl lUBJi lUrSJl of the l i n e Q^. We also assume that, i n i t i a l l y , the resonant frequency of each 2 1 1 c i r c u i t i s the same, i . e . co = - — — = j—jr~- We obtain a lumped inductance 1 1 2 2 r e l a t e d to the tube current as 4<W'> L = 2 H (9.29) TUBE and a lumped capacitance as c 2 = r^V <9-30> o The s e r i e s resistance i s found from R2 = ^ n r ( 9 - 3 1 ) O 2 2 Furthermore we require the input resistance to match to be equal to (f = f ) h - > w - 5^  This condition f i x e s the mutual inductance as M " = / ( R i " V ^ ( 5 - 3 3 ) The same procedure f o r f i n d i n g R^, C^, L^, M"' i s applied i f the terminating load (resonator) consists of two Dees. It i s to'be noted that d i f f e r e n t values r e s u l t not only for the parameters of the l i n e but also for The f a c t that the l i n e has an i n i t i a l frequency f i s due to the equality - 82 -of the e l e c t r i c and magnetic energies i n the l i n e when the loop s e l f -inductance i s taken to be part of the l i n e (see Appendix A). The parameters C 2, L 2 , R2, M " are computed by the program RESLINE. 9.4 Representation of the Whole System i n Terms of Lumped Parameters Since i t was desired to know the behaviour of resonator parameters i . e . the resonator voltage, phase, power loss etc. only i n the v i c i n i t y of resonance, the two Dees and the resonant l i n e were represented with lumped parameters. A lumped parameter representation was also necessary when c a l c u l a t i n g the transient response of the resonator voltage and the i n t e r a c t i o n of the beam with the c a v i t y RF f i e l d s . Comparison of the r e s u l t s based on lumped parameter representation with those obtained e i t h e r from d i r e c t measurements or by the program MATCH shows to what extent the representation i s v a l i d (see Section IV.1.8). The representation of the system i s shown i n F i g . 43(b). The transfer matrix could be obtained i n a s i m i l a r way as i n Section II.10.1. Instead the c i r c u i t was solved by w r i t i n g down the Kirchhoff's equations and c a l c u l a t i n g i n d i v i d u a l voltages and currents which were of p a r t i c u l a r i n t e r e s t . The input impedance was f i r s t matched to the tube resistance, the operating frequency was then v a r i e d and the voltages and currents computed. The input impedance and the resonator voltage as a function of the generator frequency are plotted i n Figs. 39, 40 (CRM). The i n v e s t i g a t i o n that has been ca r r i e d out with regard to the equivalence of the lumped parameter representation with our r e a l system has indi c a t e d that the whole system possesses the properties of a simple p a r a l l e l resonant c i r c u i t with a bandwidth almost equal to the bandwidth of the resonator (Figs. 39, 40). Since a proper match exi s t s between the resonator and the tube the loaded q u a l i t y f a c t o r i s equal to one h a l f the unloaded one. For t h i s compare again the bandwidths i n F i g s . 39, 40. - 83 -10. RESONANT OPERATION OF THE RF SYSTEM 10.1 Operation at the Maximum Power Transfer When matching the resonant load two conditions must be s a t i s f i e d . F i r s t we have to provide a matching network with such parameters that a proper tube-resonator voltage step-up r a t i o can be achieved. Secondly we want to obtain a match at a maximum power tr a n s f e r . The tube w i l l then operate with maximum e f f i c i e n c y . Two d i f f e r e n t methods e x i s t which may be applied to designing the matching network. The f i r s t one we are to deal with assumes that the generator frequency i s tuned p r e c i s e l y to the resonant frequency of the resonator. At resonance, the resonator presents a pure r e s i s t i v e impedance. This shunt resistance i s to be matched to the tube operating resistance by means of a resonant l i n e . To s i m p l i f y tuning of the whole system, three capacitors connected at various points along the transmission l i n e are at our disp o s a l . The whole system i s shown i n F i g . 41. The resonator and the l i n e are characterized by d i s t r i b u t e d parameters. The resonator voltage and current are rela t e d to the tube voltage and current by the following set of matrices: T = B 0 (10.1) T = 2 D cosh Y J & J Zj sinh Y ^ j — s i n h Y I zl ' l 1 cosh Y 1 1 (10.2) j 0 3 C 5 1 (10.3) - 84 -cosh y„l " 2 2 -rsinh Y ft Z 2 2 2 Z sinh Y ft 2 " 2 2 cosh v £ ' 2 2 ( 1 0 . 4 ) T 5 = A B 5 5 C R D 5 5 j u ^ 1 ( 1 0 . 5 ) T = 6 • D cosh Y £ „ 3 3 ^ - s i n h y 3 £ 3 Z 3 sinh cosh Y A ' 3 3 ( 1 0 . 6 ) T = 7 . A B 7 7 C 7 °7 ( 1 0 . 7 ) T = A B 0 - j CJM" ( 1 0 . 8 ) T = 9 cosh Y ^ 1h sinh Y l f \ ^ i n h Y [ + ^ cosh Y ft ( 1 0 . 9 ) T = 1 0 A i o B i o c i o D i o ( 1 0 . 1 0 ) - 8 5 "11 A l l B l l C D 11 11 cosh Y £ 5 5 f-sinh y 5 £ 5 Z sinh Y & 5 5 5 cosh Y 5^ 5 ( 1 0 . 1 1 ) We know that t h i s represents a set of two terminal p a i r networks i n cascade. The tran s f e r matrix of such a system i s an ordered product of a l l the i n d i v i d u a l matrices 1 1 T = TT T. i = l 1 I t then follows ( 1 0 . 1 2 ) or I T = T T I T L J ( 1 0 . 1 3 ) 9 TT T. i = l ( 1 0 . 1 4 ) where V T , 1^ are the voltage and current at the root of the resonator, V , 1^ are the voltage and current at the hot arm t i p i n the resonator, V J , I J are the voltage and current at the transmission l i n e input. M' i s given by eqn. ( 9 . 1 7 ) and i s given by eqn. ( 1 0 . 2 0 ) . £^ are lentghs of the f i r s t Dee, of the second Dee, and of the transmission l i n e sections. i s the complex propagation constant. The input impedance i s determined from z i = T ( 1 0 . 1 5 ) Matching the resonator shunt impedance to the tube operating impedance i s the primary task of the computer code RESLINE. Once the tube parameters have been f i x e d , the parameters of the resonant l i n e and the capacitors are varied u n t i l - 86 -the desired input impedance i s attained. The maximum power transfer occurs when Z, UBE (10.16) Besides the parameters necessary to achieve a match the program computes a number of other parameters at various points i n the system. The program computes the q u a l i t y f a c t o r s , power losses, voltages, currents, impedances, phases, stored energies, lumped constant equivalents, standing wave r a t i o s and a maximum voltage along the l i n e . A complete l i s t of a l l parameters computed by the program can be found i n Appendix A. We w i l l now describe a second method which assumes that the generator frequency i s d i f f e r e n t from the resonator resonant frequency. Let us consider a resonator which consists of one Dee [Fig. 42(a)]. Representing a Dee as a p a r a l l e l resonant c i r c u i t with lumped constants Rp, L, C we f i n d that the impedance measured across the terminals DD' i s co0 03 (10.17) Transforming t h i s impedance i n order to obtain the impedance seen by the matching network at a coupling point (terminals BB') r e s u l t s i n 2 = T~2- 772" k w0 1 032 R P - J R P Q CO C0o 1 + <f co cog wo co (10.18) where k, the resonator voltage step-up r a t i o , i s defined i n Section II.9.2. In the neighbourhood of f the re a c t i v e component of Z" i s small. We replace Z' by (10.19) with too (10.20) - 87 -to 2 (10.21) o In the v i c i n i t y of f the system can be represented as shown i n F i g . 42(b), where V i s the resonator peak voltage, V„,TTT>1:, i s the tube voltage, L T i s the 1 lUrSrL Li loop selfinductance, C g i s the capacitance that provides a proper voltage transformation r a t i o , V i s the loop induced voltage (jtoL I not included). In order to meet the f i r s t condition, i . e . to achieve a proper voltage transformation r a t i o we must have xc + h V = V — (10.22) L TUBE X U J We note that for | v„ T T 1 JJ < | V j the capacitor C would have to be replaced by an TUBE L S inductance. Now for given values of V , V , X = toL the value, of 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 l e f t with the only unknown that i s found from the second condition ( r e s i s t i v e input impedance) [Re ( Z ' ) ] 2 + [ \ + Imag (Z')][X C + ^ + Imag (Z')] = 0 (10.23) The phase s h i f t between the resonator voltage and the tube voltage i s obtained as follows. Applying Thevenin's theorem we s i m p l i f y our c i r c u i t and we f i n d -1 X F 6 = tan - I r (10.24) where To enable a transf e r of power from the tube to the Dees a resonant l i n e could be used. The length of the l i n e must be a multiple of h a l f wavelength. The complete c i r c u i t looks as drawn i n Fig.42(b). 88 -10.2 Other Resonances of the System Normally a section of transmission l i n e , loaded or unloaded, resonates at a frequency given by i t s p h y s i c a l length and a foreshortening and approximate-l y at a l l integer multiples of t h i s fundamental frequency. I t was suspected that t h i s might not hold i f we placed several capacitors at d i f f e r e n t points along the l i n e . The primary task of the computer code MATCH was to inv e s t i g a t e whether the TRIUMF resonator- transmission l i n e system could resonate at frequencies d i f f e r e n t from f for which the main program RESLINE fixed the necessary parameters. The two Dees were represented with lumped parameters as two p a r a l l e l resonant c i r c u i t s . The resonant l i n e was represented with d i s t r i b u t e d parameters. The system co n s i s t i n g of the resonator and the transmission l i n e i s said to be i n resonance i f the phase of the input impedance i s equal to zero. Finding the input impedance [see F i g . 43(a)] i s the f i r s t operation of the program. Once again we describe t h i s operation by the following set of matrices: T^, T^, .... , T ^ . T^, T^, T' = T " = T" = 9 eqns. (10.1), • • • y (10.6). t k' 7 B ' 7 M jto(L 2Lg M C7 D 7 - j L 2 M A 8 B' 8 1 0 C 8 D 8 1 R 2 1 o O J > A' 9 B' 9 1 0 D' 9 JwC 2 1 (10.25) (10.26) (10.27) - 89 -1 0 1 0 10 B i o D ; 1 0 (10.28) "11 A ' B ' 1 1 1 1 1 1 11 jcoCj 1 (10.29) 1 2 1 2 12 12 12 1 R. (10.30) '13 1 3 13 1 3 D' 1 3 1 1 (10.31) The transf e r matrix i s given by a product 13 Ul 1 The mutual inductance M i s given by eqn. (9.13). We can write VT I L l = T" T I I « > L l (10.32) (10.33) The input impedance Z^ . i s given by eqn. (10.15). Note that i n the v i c i n i t y of each of the resonant frequencies of the resonator, d i f f e r e n t lumped parameters of the resonator must be used. In the program, the impedance seen at the end of the l i n e i s obtained by transforming the resonator impedance using e i t h e r a voltage step-down r a t i o or a mutual - 90 -inductance. Although k was defined d i f f e r e n t l y than M, the dif f e r e n c e i n transformed impedances using e i t h e r k or M i s vanishingly small i n the frequency range of i n t e r e s t . The following conclusions were drawn from the re s u l t s of the program MATCH. i ) The system consisting of eit h e r one Dee or two Dees and a resonant transmission l i n e has two or four resonant frequencies, r e s p e c t i v e l y . Other computed resonances are associated with a coupling loop, i i ) One of the computed resonant frequencies of the system i s equal to the frequency f at which the l i n e parameters were adjusted to y i e l d a desired input impedance. The system i s matched for a maximum power transfer only at th i s frequency, i i i ) Around f Q , the system behaves l i k e a simple p a r a l l e l resonant c i r c u i t . The whole system must, therefore, be tuned at the resonant frequency f of the resonator. The input impedance seen by the tube i s plotted i n F i g . 39. The computed resonant frequencies of the system are presented i n Table XX. The resonant properties alternate between those of a series resonant c i r c u i t and those of a p a r a l l e l resonant c i r c u i t . The program MATCH also computes voltages, currents, power losses, q u a l i t y f a c t o r , impedances and phases i n the l i n e . The program can accept e i t h e r a non-resonant or i n d u c t i v e l y coupled and c a p a c i t i v e l y coupled resonant load. Three capacitors may be connected anywhere along the l i n e . - 91 -10.3 Possible Operating Conditions If the RF system i s excited at a frequency equal to the resonator resonant frequency the two Dees present a pure r e s i s t i v e load at the coupling point. Only for t h i s frequency are the resonator t i p to loop voltage r a t i o and the voltage and current d i s t r i b u t i o n s i n the l i n e close to the computed values. The voltage node p o s i t i o n should then be found as indicated by the program. Once the resonator resonant frequency i s known the l i n e should be tuned at t h i s frequency to obtain the correct input impedance. However,the system can also be run at a frequency which i s s l i g h t l y d i f f e r e n t from the resonator resonant frequency. In the v i c i n i t y of the resonator resonant frequency we can represent the system as shown i n F i g . 42(c). The expressions for X^, R^ for a two Dee resonator are complicated. However, for a one Dee resonator they are given by eqns. (10.20) and (10.21) from which one finds the range of frequencies f o r which the l i n e with connected capacitors, loop reactance and the resonator reactance creates a IT network which transforms R^ into R ^ ^ . I f the l i n e parameters (capacitors included) were f i x e d , there would be j u s t one frequency for which a proper match ( a desired input resistance) could be obtained. Since our capacitors are v a r i a b l e and because jwL^ and jX^ are frequency dependent, the condition to be s a t i s f i e d to produce a r e s i s t i v e input impedance i s [Re ( Z - ) ] 2 + [ \ + \ i m + Imag ( Z ' ) ] ^ + X ^ ^ + X c + Imag (Z')]=0 Under these conditions the transmission l i n e and the resonator no longer possess the same resonant frequency. The resonator t i p to loop voltage r a t i o , V /VJ*, may be e i t h e r higher or lower depending on the operating frequency. Consequently, i t i s p r i m a r i l y the voltage and current d i s t r i b u t i o n s i n the l i n e that are a f f e c t e d . For lower values of t h i s r a t i o the voltage i n the l i n e may e a s i l y be higher by a factor of two than the computed values. In t h i s case, - 92 -the whole system i s i n resonance but at the frequency f ^ f . Power can s t i l l be fed into the resonator but the losses i n and near the coupling loop may considerably increase. The tube w i l l , however, continue to feed the same power into the system as long as the tube sees the same r e s i s t i v e input impedance. For many reasons such as temperature transients i n the resonator panels, cooling water pressure v a r i a t i o n s , multipactoring etc., i t i s not anticipated that the RF system w i l l be operated at a f i x e d frequency at the beginning. The frequency s h i f t s due to temperature transients are too large to be compensated for by means of tuning bellows. The resonator unloaded frequency can be measured p r i o r to high voltage tests using a loose capacitive coupling. The l i n e i s then connected to the resonator and the system c o n s i s t i n g of two Dees and a resonant l i n e w i l l be tuned under cold conditions i n order to a t t a i n a desired r e s i s t i v e impedance at the input of l i n e at the resonator resonant frequency. The actual tuning w i l l be accomplished by means of three capacitors inserted at three d i f f e r e n t places along the l i n e . The resonator t i p to loop voltage r a t i o or the phase s h i f t between the resonator and loop currents can be taken as a reference to make sure that the tuning i s being done at the resonator frequency. Figs. 44, 45 show the resonator t i p to loop voltage r a t i o and the phase between the resonator and loop currents as functions of d r i v i n g frequency f o r a fixed resonator frequency (and vice versa). Once the tuning of the system has been f i n i s h e d the RF power can be delivered to the Dees. In order to overcome multipactoring, at the s t a r t , the main RF a m p l i f i e r w i l l be pulsed (pulses of about 1 msec width and up to 140 kV amplitude). This pulsing of the amplifier w i l l be stopped once the resonator voltage has reached about 20 kV, at which moment a switch to a s e l f -o s c i l l a t o r y mode occurs (driving s i g n a l supplied by the resonator). The resonator frequency w i l l change owing to temperature transients, cooling water pressure v a r i a t i o n s and the RF forces. The operating frequency w i l l then 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 around the whole feedback loop i s always maintained equal to 360 deg (or 0 deg). At t h i s new operating frequency which i s now s l i g h t l y d i f f e r e n t from the resonator frequency the phase s h i f t introduced by the resonator i s compensated by an opposite phase s h i f t introduced by the other RF components. Computer r e s u l t s have shown that the d r i f t i n the resonator frequency as large as -30 kHz would lead to a difference between the resonator frequency and the operating frequency of at most -0.5 kHz (CRM). This frequency d i f f e r e n c e i s small and would not a f f e c t the voltage d i s t r i b u t i o n i n the f i r s t section of l i n e (compared to computed values f o r f = f ) provided that the i n i t i a l tuning of the l i n e i n order to obtain a r e s i s t i v e input impedance was done at the resonator frequency.f . Should the r e s i s t i v e input impedance be attained by tuning the l i n e at the frequency which was d i f f e r e n t by more than Af = f - f = ±1 kHz from the resonator frequency the voltage d i s t r i b u t i o n i n the f i r s t s e c tion of the l i n e would be s i g n i f i c a n t l y changed. In addition to t h i s , the d r i f t i n the resonator frequency would now lead to e i t h e r a decrease or an increase of Af = f - f and, consequently, the voltage l e v e l i n the f i r s t section would eit h e r go up or down. I t should be understood that the d r i f t i n the resonator frequency can be considered as a second order e f f e c t and may be of s i g n i f i c a n c e only i f the tuning of the transmission l i n e was c a r r i e d out at a frequency very far from the resonator frequency or 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 i n the resonator frequency ceases, a switch from the s e l f -o s c i l l a t o r y mode to the f i x e d frequency mode can take place. Any sudden d r i f t s i n resonator frequency (due to a reactive beam load and mechanical o s c i l l a t i o n s ) during the f i x e d frequency mode w i l l be corrected by means of - 94 -tuning bellows. Uncompensated resonator frequency s h i f t s would require an increase i n the power input i n order to keep the resonator peak voltage at a given l e v e l . Both the power input increase and the differ e n c e between the resonator frequency and the operating frequency could influence the voltage and current l e v e l s i n the f i r s t section of the l i n e . - 95 -CHAPTER I I I . EXPERIMENTAL TESTS To reduce the cost and space requirements most model measurements were done at h a l f - s c a l e . The tests were c a r r i e d out both at medium and low power l e v e l s with combinations consisting of various numbers of resonant sections. By a medium power l e v e l one means that voltages of about 500 V were reached on cavity while a low power l e v e l represents voltages of a few V o l t s . The segments were made up of copper covered plywood. Insulators (polyesterene) we-'re employed to hold the i n d i v i d u a l panels i n a correct p o s i t i o n and to exclude any mechanical o s c i l l a t i o n s . To ensure good contacts everywhere the segments were bolted at the root i n both the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . A l l segments were also held together r i g i d l y near the t i p . Since a l l dimensions were scaled down by a factor of two the resonator c h a r a c t e r i s t i c impedance remained constant. The length and the t i p load-ing capacitance were reduced by a factor of two. This resulted i n a doubled frequency, higher power loss and lower q u a l i t y factor compared with a f u l l -s cale resonator, see Sections II.1.6. The contacts between the hot arm t i p s were provided by several cm long copper s t r i p s soldered to the hot arms. It was found that the hot arm t i p - t o -t i p contacts influenced the e l e c t r i c a l properties of the cav i t y . For example, the q u a l i t y f a c t o r of the resonator with no contacts between the hot arm t i p s dropped by a fa c t o r of f i v e . This was mainly due to a fl u x leakage through the gaps between the panels. It was also found that the voltage uniformity along the accelerating 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 connecting s t r i p s across the gaps between the resonator segments the better was the voltage uniformity along the accelerating gap during the model measurements. The q u a l i t y factors were always measured with a very loose capacitive coupling between the resonator and the generator. The measurements at low and - 96 -medium power l e v e l s showed no di f f e r e n c e i n the qu a l i t y f a c t o r . The values of the Q's measured on i n d i v i d u a l resonators are presented i n the next Sections. The computed values are i n brackets. Capacitive probes made up of s o l i d state diodes were employed to measure r e l a t i v e voltages i n the upper resonator segments. The probes were c a l i b r a t e d p r i o r to any measurements by means of a thermionic diode. During the tests a l l voltages were r e l a t e d to a reference probe voltage and r e l a t i v e voltage differences were recorded. The e f f e c t of the capacitive probes on the Q ( a drop by about 5% ) was small and, therefore, neglected. MEASUREMENT OF RESONATOR PARAMETERS A. Resonant frequency In general, e s t a b l i s h i n g the resonant frequency according to eqn. (1.5) presented no problems. However, some inaccuracy might be encountered when one or two sections are assembled. The e f f e c t of f l u x guides on the c h a r a c t e r i s t i c impedance per unit width of the resonator i s very large. B. Tip loading capacity The value of t h i s capacity was determined from the equation (1.5). For a resonator co n s i s t i n g of 20 sections i t i s possible to c a l c u l a t e Z q with a good accuracy. If we now measure the resonator length and the resonant frequency we can e a s i l y evaluate C^p* A s u b s t i t u t i o n of measured values i n eqn. (1.5) resulted i n C j . p = 7.5 p'F per resonator segment at h a l f - s c a l e . This would mean that a f u l l - s c a l e value i s C T Ip = 15 pF. The l a t e s t measurements on CRM resonators ind i c a t e that the measured value i s around 13 pF, the exact value depending on the nature of the beam probe housing i n the CRM tank. C. Quality factors As the value for the q u a l i t y f a c t o r i s d i r e c t l y proportional to the c h a r a c t e r i s t i c impedance any inaccuracy i n Z ^ r e f l e c t s i n the value of the Q. The e f f e c t of varying resonator parameters on the value of the q u a l i t y f a c t o r i s shown i n F i g . 7. The measured q u a l i t y factors varied depending upon the contacts at the root and between the hot arm t i p s . C e r t a i n l y the main factor i n f l u e n c i n g the Q was a mechanical misalignment of i n d i v i d u a l segments. The segments tuned to s l i g h t l y d i f f e r e n t resonant frequencies cause a drop i n the o v e r a l l Q. The q u a l i t y factors that were measured on h a l f - s c a l e resonators with good e l e c t r i c a l contacts and mechanical alignment of i n d i v i d u a l segments were 4600 (5500) and 6100 (9500) for the fundamental and the t h i r d harmonic, re s p e c t i v e l y . The q u a l i t y factors measured on a f u l l - s c a l e s i n g l e section model amounted to 6000 (7200) and 9300 (12500). - 98 -D. Phase measurements The phase tests at h a l f - s c a l e were aimed at f i n d i n g out what the voltage phase di f f e r e n c e between the two extreme ends of a Dee and across the accelerating gap was. The RF voltage phase di f f e r e n c e measured on a resonator made up of 20 sections was found to be l e s s than 1 deg. Also the RF phase differe n c e between the upper and lower row of resonator segments did not exceed 1 deg. The phase di f f e r e n c e across the accelerating gap when the two Dees made up of 10 sections each were assembled amounted to 180 ± 0.5 deg and was neither affected by a hot arm d e f l e c t i o n nor by s h i f t i n g the root plunger. However, any leakage of the RF energy from the accelerating gap had to be avoided. The tests were done at a medium power l e v e l with coupling to one Dee only. 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 decided to excite the resonator system by means of a s i n g l e coupling loop. E. Resonator lumped capacity The resonator lumped capacity was determined by i n s e r t i n g a capacitor of a known value at the high voltage end of the resonator and measuring the new resonant frequency. Given the o r i g i n a l resonant frequency f Q , the new resonant frequency f^ and the a d d i t i o n a l capacity AC, the lumped capacity of the resonator i s calculated from C = M <w2-1 The te s t was done with a two section resonator modelled at h a l f - s c a l e . Four capacitors, approximately 1 pF each, were connected between the hot arm t i p s and the grounding p l a t e . The measured values of f Q , f , AC were: AC = 4.3 pF, f = 45.7146 MHz, f = 45.3803 MHz, Z = 36.4 ft, I = 1.5619 m, C - = 7 pF. o 1 o TIP r After s u b s t i t u t i o n above the value of C = 291 pF was obtained. The 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 capacity resulted i n C = 302 pF. The measured value was within 5% of the computed one. - 99 -2. FREQUENCY TUNING 2.1 Tuning Stub It i s w e l l known that the resultant frequencies of two coupled c i r c u i t s can be influenced by varying the parameters of both c i r c u i t s simultaneously or by changing the parameters of one c i r c u i t only. The same deductions hold i f we have two resonant c a v i t i e s coupled together. Let us assume that i n place of the other resonator we introduce a resonant l i n e , approximately a quarter wavelength long, short c i r c u i t e d at the far end and coupled either c a p a c i t i v e l y or i n d u c t i v e l y to the resonator. The simplest way to tune the system con s i s t i n g of the resonator and a resonant tuning stub i s to vary the p o s i t i o n of the tuning stub plunger. Any such change represents a change i n both a lumped capacity and inductance of the tuning stub (Fig. 59). Consequently the resonant frequency of the system i s a l t e r e d . In our system the magnitude of the frequency v a r i a t i o n also depends on the selected coupling point. For the tests several resonators were modelled at h a l f - s c a l e . The main aim of these tests was to i ) investigate the range of the resonator tuning by means of tuning stubs and i i ) f i n d a voltage v a r i a t i o n along the accelerating gap caused by s h i f t i n g a tuning stub plunger. The tests also included a measurement of the q u a l i t y factors of the c a v i t y -stub system. The r e s u l t s obtained with a ten section model are plo t t e d i n Figs. 46, 47. Four stubs coupled c a p a c i t i v e l y were connected to the resonator i n sections #3 and #8 (two per upper row, two per lower row of resonator segments). The R ST c h a r a c t e r i s t i c impedances of the system were Z = 4 ft,Z =43.6ft. The J o o magnitude of the coupling capacitance was C = 18 pF. The voltage i n a - 100 -p a r t i c u l a r section was subject to two e f f e c t s . F i r s t the resonant frequency of t h i s system was dependent on the stub shorting plunger p o s i t i o n . Secondly due to the fac t that the coupling was not uniformly d i s t r i b u t e d along the accelerating gap the f i e l d i n the resonator i n the proximity of the tuning stub was d i s t o r t e d . Capacitive pick-up probes made up of s o l i d state diodes measured voltages with respect to a reference probe voltage i n section #4. A l l probes were c a l i b r a t e d p r i o r to measurements. Graphs i n F i g . 46 show the frequency and q u a l i t y factor v a r i a t i o n s measured on a f i v e section resonator. The stub was coupled c a p a c i t i v e l y i n section #3. The parameters were: R ST Z q = 4.4 ft, Z q = 43.6 ft, C^ = 10.2 pF. F i g . 46 also shows a frequency v a r i a t i o n measured on a twenty section model (each Dee made up of 10 sec t i o n s ) . Two stubs coupled c a p a c i t i v e l y were connected i n each Dee i n section #9, top and bottom. A corresponding voltage v a r i a t i o n i s shown i n F i g . 47. The R ST following parameters describe the system: Z q = 4 ft, Z q = 43.6 ft, C^ = 11 pF. In a l l tests mentioned, the coupling point was near the high voltage end of the resonator. The terminating capacitance C^ , was produced by the p l a t e - t o -hot arm capacitance. I t was deduced that the voltage v a r i a t i o n could not be held within the required tolerances and moreover, the qu a l i t y factor showed a s i g n i f i c a n t drop. In the case of a push-pull mode i n the resonator-stub system a very high voltage would develop at a coupling point. 2.2 Ground Arm De f l e c t i o n This kind of tuning was investigated on a f i v e section model assembled at h a l f - s c a l e . The hinged ends [see F i g . 15(b)] of both the upper and lower ground arms, i . e . the plates of r e a l dimensions 12 i n . x 16 i n . , could be moved i n so that a d e f l e c t i o n up to Ag = - 1.5 cm at the t i p was produced. The max frequency v a r i a t i o n due to the deflected ground arm t i p was quite large and we conclude that t h i s method of tuning could be used both f o r f i n e and coarse - 101 -frequency tuning. The measured values, p l o t t e d i n F i g . 18(a), were i n good agreement with the computed ones. Discrepancies between computed and measured values f o r larger d e f l e c t i o n s may be accounted for a change i n the hot arm-to-grounding plate capacitance which was not included i n c a l c u l a t i o n . The q u a l i t y f a c tors were unaffected by the d e f l e c t i o n of the ground arm t i p s [see F i g . 18(b)]. The corresponding voltage v a r i a t i o n i s shown i n F i g . 49. I f a l l the resonant sections were aligned properly and tuned thus to the same frequency the measured voltages i n d i f f e r e n t sections should always give the same value for any d e f l e c t i o n of the ground arm t i p s . D i f f e r e n t values of voltage i n our case i n d i c a t e that the mechanical alignment of i n d i v i d u a l segments was not p e r f e c t . 2.3 C y l i n d r i c a l Capacitors Eight c y l i n d r i c a l capacitors [see F i g . 15(c)], 4 i n . i n diameter, were i n s t a l l e d i n a ten section resonator i n both upper and lower resonator segments #4, #5, #6, #7. Since the a d d i t i o n a l capacitance produced by moving the copper cylinders inwards was not spread uniformly along the accelerating gap the f i e l d near the gap was d i s t o r t e d to a c e r t a i n extent. I t was also expected that a drop i n q u a l i t y factors would occur as a r e s u l t of d i s t o r t e d equipotentials. The frequency v a r i a t i o n i s shown i n F i g . 50(a), while F i g . 50(b) shows how the q u a l i t y factors were affected by changing the p o s i t i o n of these capacitors. The voltage v a r i a t i o n exceeded the permissible l i m i t s (see F i g . 51). The conclusion was that t h i s kind of tuning could not be employed. The resonator was characterized by the following parameters: Z q = 43.2 ft, £ = 1.55 m, CTIP " 7 P F ' - 102 -2.4 Capacitive Plates An i n v e s t i g a t i o n was c a r r i e d out on a ten section model constructed at h a l f - s c a l e . Capacitive plates made up from copper covered plywood, of r e a l dimensions 4 m x 0.11 m x 0.01 m, were mounted on the ground arm t i p s i n a manner that permitted the t i p loading capacity to be increased [see Fig.15(a)] by moving the plates down towards the hot arm t i p s . The actual contact between the plates and the ground arms was achieved by means of copper s t r i p s , 4 per resonator segment, however, to avoid any d i s t o r t i o n of the f i e l d a continuous contact with the ground arms should have been made. The parameters of the system are presented i n Section III.2.3. The measured frequency change i s shown i n F i g . 16, which shows reasonable agreement with the computed values. Once again the e f f e c t of capacitive plates on the tip-to-grounded plate capacity was not included i n c a l c u l a t i o n . A percentage voltage d i f f e r e n c e as measured and r e l a t e d to a reference probe i n section #2 i s plotted i n F i g . 52. 2.5 Inductive Loops at the Root Several models were assembled at h a l f - s c a l e to investigate tuning of the cavity by means of r o t a t i n g loops and f i n s as shown i n F i g . 15(d). If the f i n s , eventually, the loops, are not too f a r apart we can vary the e f f e c t i v e length of the resonator by r o t a t i n g e i t h e r the loops or the f i n s with respect to t h e i r axis. The e f f e c t on the resonant frequency due to a d i s t o r t i o n of the f i e l d near the root i s equivalent to the e f f e c t caused by s h i f t i n g the root plane. The frequency v a r i a t i o n measured on a three section resonator at h a l f -scale i s shown i n F i g . 53. Sixteen f i n s , 0.5 i n . x 1.5 i n . x 2.5 i n . , were mounted i n the middle segments.. A l l the f i n s were gradually turned 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 . In the next t r i a l the f i n s #3 and #6 were rotated (see F i g . 53). The q u a l i t y factor of an unperturbed cavity was = 4560. It can be seen from the graphs i n F i g . 53 that the q u a l i t y factor - 103 -i s affected by the i n s e r t i o n of f i n s i n s i d e the cavity and further a l t e r e d i f they are rotated. This i s due to high losses i n the f i n s due to the high magnetic f i e l d density near the root and the consequent huge induced currents i n the f i n s . The second set of experiments was done with a two section resonator where e i t h e r f i n s or loops were mounted on to the root plungers. There were eight f i n s per each segment. The q u a l i t y factors of the resonator without tuning elements were 4250 and 5950 f o r the f i r s t and t h i r d harmonic, re s p e c t i v e l y . Both the loops and the f i n s were made up of 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 . thick copper. The re s u l t s are p l o t t e d i n Figs. 54, 55. 2.6 Third Harmonic Tuning Diaphragms The measurements done at h a l f - s c a l e included p r a c t i c a l tests with two frequencies i n j e c t e d into the resonator and were to prove that a "square wave" could be obtained without great d i f f i c u l t y . Using two 50 PJ coaxial transmission l i n e s and e x c i t i n g both harmonics separately by means of separate coupling loops, the resonator could be operated at various mixtures of the fundamental and the t h i r d harmonic where the t r i p l e r was used. Though the need f o r an a d d i t i o n a l tuning element i n order to obtain a proper frequency r a t i o was envisaged, no tuning diaphragms were used at t h i s time. The whole system consisted of several coupled c i r c u i t s and the resultant resonant frequencies f , f^ were the frequencies of the ent i r e system. By means of feedback v i a the transmission l i n e s the state could be found when the r a t i o f g / f ^ = 3 and the desired amount of the t h i r d harmonic was present. Good phase s t a b i l i t y was attained 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 the two Dees, each made up of 5 sections, i n push-pull mode with coupling to one Dee only there was no d i f f i c u l t y i n obtaining the "square wave",as w e l l . The amount of the t h i r d harmonic could be varied from 0 to 20% - 104 -of the fundamental l e v e l of 500 V. Several problems were not solved. The natural t h i r d harmonic resonator frequency f Q ^ was not three times f Q . To achieve the desired cavity voltage wave form the fundamental a m p l i f i e r was operated at a frequency s l i g h t l y lower than f . The t h i r d harmonic amplifier was operated at f ° 1 03 These problems were solved with a s i n g l e section f u l l - s c a l e resonator o r i g i n a l l y designed to v e r i f y the RF system program RESLINE. This resonator, made i n f u l l - s c a l e , operated at the nominal frequency 22.66 MHz of the fundamental (see Section I I I . 5 ) . As tuning elements, two diaphragms, 0.25 i n . x 2.5 i n . x 25 i n . , were chosen and connected to the ground arms at the p o s i t i o n 5X^/24 = 108.5 i n . from the root (see F i g . 56). The q u a l i t y factors of 5400 and 6100 were obtained, r e s p e c t i v e l y . The mechanical construction of the diaphragms made i t possible to turn 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 p o s i t i o n . The frequency v a r i a t i o n as measured with the diaphragms turned gradually 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 order of magnitude greater than 1 6 the values predicted using the perturbation theory developed by Slater [see Figs. 21 and 57(a)]. The same measurement repeated for the t h i r d harmonic proved that the fundamental resonant frequency was influenced more than the t h i r d harmonic resonant frequency, but the same disagreement with c a l c u l a t i o n was found as f a r as the percentage change was concerned. However, the measured maximum detuning was i n good agreement with the values calculated using a d i f f e r e n t method of perturbation theory, namely considering a capacitive e f f e c t produced by a tuning diaphragm (see F i g . 22). The r a t i o f / f vs p o s i t i o n of diaphragms was plotted i n F i g . 57(b) and t h i s indicated the p o s i t i o n where the r a t i o was exactly equal to 3. A s i n g l e o s c i l l a t o r was used as a s i g n a l source, the t h i r d harmonic frequency being generated by a t r i p l e r . F i g . 56(a) shows the arrangement of - 105 -the experiment at the time when the resonator was run with the two harmonics. Both harmonics were injec t e d at about the same distance from the root, the sensor being a capacitive probe at the fro n t . The t h i r d harmonic amplitude varied from 0 to 20% at the 50 V l e v e l of the fundamental. Quality factors remained nearly constant during motion of the diaphragms. At t h i s stage the fundamental frequency was changed by +3.2% ( s h i f t i n g the root plane) and the resonator was retuned so that f / f was again equal °3 °i to 3. The diaphragms were turned by about 20 deg to retune the resonator to both frequencies. As the measured frequency v a r i a t i o n did not agree with the r e s u l t s based on the perturbation theory which considered a change i n the e l e c t r i c and magnetic energies stored i n the resonator i t was concluded that to get better understanding of what was happening the p o s i t i o n of the diaphragms should be alte r e d . The two diaphragms were now mounted i n the middle of resonator, X /8 from the root. The resonant frequency v a r i a t i o n and the r a t i o f / f oi 03' 01 may be seen i n F i g . 58. The r a t i o never exceeded 3 staying constant over a range of diaphragm motion. A further change i n p o s i t i o n of the diaphragms to X I l k from the root and s i m i l a r investigations as before resulted i n curves 01 shown i n F i g . 58. The r e s u l t s were unsatisfactory since the r a t i o , again, was always below 3. I t was concluded that the perturbation was too b i g , and that a capacitive e f f e c t predominated, the new resonant frequency being mainly determined by a change i n the t o t a l lumped capacitance. Table XIII summarizes the maximum frequency change produced by moving the diaphragms 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 . The measured values are compared with computed ones. The low frequency capacitance increase AC^ produced by turning the diaphragm 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 measured and found to be equal to 6 pF. - 106 -TABLE XIII Computed and measured frequency s h i f t s caused by turning the diaphragms 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 P o s i t i o n from root (cm) Af o i measured (%) Af oi computed (%) Af 03 measured (%) Af 03 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 - 107 -3. VOLTAGE AND FREQUENCY VARIATIONS DUE TO MECHANICAL MISALIGNMENTS Since each Dee consists of 40 resonator segments quite severe mechanical tolerances must be met i n order to achieve good e l e c t r i c a l p roperties. A misalignment of e i t h e r resonator panels or root plungers 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, resonator length and a t i p loading capacity. This leads to a non-uniform voltage d i s t r i b u t i o n along the accelerating gap. The expected mechanical misalignment i n root plungers i s Ail = ±0.040 i n . , i n resonator panels ( v e r t i c a l l y ) Ag = ±0.020 i n . I t was necessary to v e r i f y on a model how large the voltage and frequency v a r i a t i o n s were i f one of the above mentioned mechanical misalignment occured. The measurements were f i r s t done on a two Dee resonator at h a l f - s c a l e where each Dee was made up of 10 sections and l a t e r on a one Dee resonator consisting of 18 sections. In the f i r s t graph, F i g . 60, the percentage voltage change along the accelerating gap vs a d e f l e c t i o n of the upper hot arm i n section #3 was r e l a t e d to a reference probe i n section //9. A corresponding frequency v a r i a t i o n i s shown i n F i g . 61. The next graph, F i g . 62, shows the voltage v a r i a t i o n produced by moving root plungers i n section #7. The measured q u a l i t y f a c t o r was Q^  = 3900 without the probes, Q-^  = 3500 with the probes i n . Similar r e s u l t s were obtained with a two Dee resonator, Fig.61, 62. The measurement gave Q = 3200. It should be noted here that any misalignment i n one Dee caused the v i r t u a l nodal plane i n the middle to s h i f t from i t s ce n t r a l p o s i t i o n . An asymmetry i n the Dee voltages immediately followed. Also any misalignment with regard to root plungers caused a s i g n i f i c a n t decrease i n the value of the t h i r d harmonic q u a l i t y f a c t o r (see F i g . 64). - 108 -4. RESONATOR MODIFICATIONS 4.1 Extreme End Segments A f i v e s ection resonator was f i r s t assembled and the q u a l i t y factors were measured and found to be 4600 (5200) and 5500 (9080) for the fundamental and the t h i r d harmonic, r e s p e c t i v e l y . Replacing the f i f t h section with a tapered one and measuring the Q's again resulted i n the low values: Q^  = 3000, Q^ = 3000. The voltage uniformity along the accelerating gap was also u n s a t i s f a c t o r y . The extreme end section was now modified i n order to obtain a constant c h a r a c t e r i s t i c impedance throughout the section (see Section II.5.1). This modification included an adjustment of the hot arms i n the extreme end section, Fi g . 23, both the ground arms and hot arm t i p s being l e f t unchanged. The voltage v a r i a t i o n along the accelerating gap was reasonably good for the fundamental. The q u a l i t y f a c t o r f o r the fundamental was improved to Q = 4200, however, the t h i r d harmonic Q remained at about Q^ = 3100. This would indi c a t e that the increase i n power loss due to a modified extreme end section represented 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 loss i n the t h i r d harmonic mode. The percentage increase i n the t h i r d harmonic power loss i s much larger than that for the fundamental, because the t h i r d harmonic has two current maxima per section while the fundamental has only one. Should a f u l l Dee made up of 20 sections be assembled, the percentage increase i n the t o t a l power loss would be lowered by a factor of two both f o r the f i r s t and t h i r d harmonics (compared with a f i v e section model). At t h i s stage a decision was made to enlarge the vacuum tank and thus accomodate a normal resonant section at e i t h e r end of a Dee. 4.2 Central Resonator Segments The measurements started on a 20 section model with no centre post and - 109 -with no cut-out. The aim was to obtain reference values for the q u a l i t y f a c tors of the unmodified model. The measurement resulted i n Q = 4000 (5250), = 2100 (9200). The value of the t h i r d harmonic q u a l i t y factor was s e n s i t i v e to any imperfections i n the root contacts and mechanical misalignment of resonator segments. Two c e n t r a l sections were then taken out and replaced with sections with modified hot arm t i p s (cut-out) and the centre post was also i n s t a l l e d . The values of Q = 4000, Q = 460 were then measured. A s i g n i f i c a n t improvement i n Q occurred as soon as the root plungers i n sections 3 with the cut-out were moved i n . The value increased to Q = 1400. At that 3 time i t was decided not to proceed with further investigations and improvements on a 20 section model and to use rather a 4 section resonator (one Dee) where any change i n mechanical construction should be immediately r e f l e c t e d i n e l e c t r i c a l properties of the resonator. I t was f e l t that by a l t e r i n g the geometry around the centre post the four sections could be tuned to the same resonant frequency. During the t e s t i n g period the dimensions of the centre post and the i n j e c t i o n gap p o s i t i o n have been alt e r e d several times to include changes demanded by the beam dynamics group. The d e t a i l e d design of the resonator i n the centre was almost 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 with respect to the accelerating gap. The measured value of the t h i r d harmonic q u a l i t y factor was Q 3 = 2500. The voltage along the t i p i n the middle section varied by about 1.5% for the t h i r d harmonic. It was also deduced from the tests that the Q was almost unaffected unless some d r a s t i c a l changes i n resonator geometry occurred. The voltage v a r i a t i o n of the fundamental along the accelerating gap i n the c e n t r a l sections was always less than .5% (see Table I ) . However, i t has been proved that a good voltage uniformity along the accelerating gap i n the c e n t r a l sections does not ne c e s s a r i l y mean that the q u a l i t y f a c t o r i s good. On the other hand a good q u a l i t y f a c t o r guarantees a good voltage uniformity. - n o -A l l the measurements were repeated once the basic design of the centre post and the i n j e c t i o n gap was f i n a l i z e d . This gap now coincides with the accelerating gap. Since the i n j e c t i o n gap i s produced by a separation between the centre post and the hot arm t i p s of one Dee, only 100 kV voltage peak w i l l be developed there. A two Dee resonator, each Dee made up of 4 sections, was assembled. F i g . 65 shows the basic geometry i n the ce n t r a l region. A l l modifications made concerned the resonator gap (hot arm tip-to-ground arm ti p ) i n the c e n t r a l resonator segments. The voltages were measured at a given g r i d of points, the d i s t r i b u t i o n of equipotentials near the accelerating gap being then obtained through use of the computer program FITVOLT. The resonant frequency and the voltage non-uniformity were then taken as a reference f o r the next modifications. The t h i r d harmonic Q could be improved to Q - 1500. - I l l -5. RESONANT LINE AS A MATCHING NETWORK The aim of the tests was to v e r i f y the computer program RESLINE. A f u l l -scale s i n g l e resonant section, modified i n order to meet the supposed c h a r a c t e r i s t i c impedance of the r e a l resonator i n the CR cyclotron, was used. This modification, i . e . a change i n a nominal s i z e of one resonator panel, was due to a d i f f e r e n t e f f e c t of the f l u x guides on the c h a r a c t e r i s t i c impedance, see Section II.1.1. Copper sheeted plywood and aluminum (for the transmission l i n e l i d ) were the only materials used. The l e f t hand part of the resonator system, i . e . the other Dee, was simulated by putting a grounded plate i n place where the v i r t u a l grounding plane goes. The resonator was excited through loose inductive or capacitive coupling. A VHF o s c i l l a t o r and a 10 W amplifier were employed as sources of RF energy. For monitoring purposes and phase measurements, simple low capacitance probes were used as RF sensors. Hewlett-Packard sampling and vector voltmeters were used with these probes. Two i d e n t i c a l DC probes representing a minimal load (< 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 voltages along the transmission l i n e by touching the inner conductor at various s t a t i o n s . The measured or computed quantities were l i n e a r l y scaled i n order to allow making comparisons. During the measurements an opposite flow of energy was obtained, i . e . from the resonator to the v i r t u a l amplifier at the input of l i n e . The RF energy was extracted from the resonator by means of the coupling loop. Under these conditions the transmission l i n e was loaded at the end by an impedance computed by the program RESLINE. The r e s i s t i v e part, i n f a c t , simulated the i n t e r n a l resistance of the power tube. I t should be emphasized that the opposite flow of energy had to be used i n order to minimize the loading e f f e c t of s o l i d state diodes on the output from the power a m p l i f i e r . The calculated values of the q u a l i t y factors (loaded and unloaded), loop self-inductance, - 112 -TABLE XIV Calculated and measured dimensions of the f u l l - s c a l e s i n g l e resonant section and the resonant l i n e Calculated Measured (cm) (cm) Hot arm length 318.60 319.02 width 62.36 62.36 t i p radius 1.91 1.91 - ground arm gap 10.16 10.16 - grounded plate gap 7.62 7.62 Ground arm length 325.62 326.64 width 82.68 82.68 - ground arm gap 38.10 38.10 Coupling loop length 44.45 44.45 width 6.35 6.35 thickness 2.54 2.54 height 4.45 4.45 p o s i t i o n 20.32 20.32 Transmission l i n e inner conductor outer conductor 1.91 x 5.12 7.09 x 10.01 1.91 x 4.74 7.09 x 10.01 length t o t a l 675.00 675.00 v e r t i c a l part 45.09 45.09 _ 113 _ c h a r a c t e r i s t i c impedance, dimensions of the resonator and the transmission l i n e are summarized i n Tables XIV, XV. In order to a r r i v e at a desired c h a r a c t e r i s t i c impedance of the transmission l i n e , the capacitance per unit length was measured and the dimensions of the inner conductor were adjusted as required. TABLE XV Calculated 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 single resonant section and the resonant l i n e Calculated Measured f o ! (MHz) 22.66 22.41 fo3 (MHz) - 67.02 Q l 7200 5970 % 12450 9300 Z o (ft) 38.3 -ZTL (ft) 50.0 50.0 L L (yH) 0.208 -Throughout the assembly period, the q u a l i t y factors of the system were measured at both the fundamental and the t h i r d harmonic frequencies (see Table XVI). The amount of power absorbed at the r e s i s t o r simulating the i n t e r n a l resistance of the tube was measured. The value of t h i s r e s i s t o r was varied, and the voltage across the r e s i s t o r was measured for each value of resistance. The capacitance connected at .this point was calculated by RESLINE, and i t was held constant during the measurements. This capacitor was - 114 -TABLE XVI Quality factor and power absorbed vs tube simulating resistance ^TUBE V P Q L (ft) (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 used to cancel the inductive component of the input impedance. At the transmission l i n e input the system can be represented as a constant current generator feeding the input resistance i n p a r a l l e l with the tube simulating resistance. The maximum power output from a constant current generator occurs when the generator conductance i s equal to the load conductance. The loaded q u a l i t y f a c t o r should then drop to one h a l f of the unloaded one. The measured voltages and corresponding losses at the end of l i n e are presented i n Table XVI. For c a l c u l a t i o n of the power absorbed, the computed value of the input resistance was used, i . e . R^ = 2083 ft. I t was concluded that the indicated value of resistance giving a maximum absorption i s i n reasonable agreement with the computed value. - 115 -An analogous measurement for the t h i r d harmonic frequency had shown that only a n e g l i g i b l e f r a c t i o n of the RF energy was flowing from the resonator into the transmission l i n e . This confirmed our expectation that the l i n e matched the resonator shunt resistance to the tube operating resistance only at the fundamental frequency. The voltages measured along the resonant transmission l i n e were i n good agreement with computed values. Both the measured and computed values for the RF fundamental are p l o t t e d i n the graph i n F i g . 66. We also investigated how the voltage and current d i s t r i b u t i o n s i n the l i n e are affected by the capacitor s e t t i n g s . The values of capacitors CP(2) and CP(3) were changed and the computer program calculated the new voltage d i s t r i b u t i o n along the l i n e and the desired tube simulating impedance. The r a t i o V T / V was kept close to the computed value by adjusting the capacitor j~ J_» CP(6) at the input of the l i n e (see F i g . 67). Larger changes i n the p o s i t i o n of the middle capacitor CP(3) were also examined (see F i g . 68). I t was proved that i t was always possible to a t t a i n a desired voltage r a t i o V / V by adjusting some parameter i n the l i n e , however, at the expense of completely d i f f e r e n t voltage and current d i s t r i b u t i o n s i n the l i n e . In order that the voltage and current d i s t r i b u t i o n s along the l i n e follow the computed values, the r a t i o of voltages across the loop and across the middle capacitor must be readjusted according to the computed value,as w e l l . This accounts for the discrepancy between Computed and measured values i n F i g s . 67 and 68. The r a t i o of voltages across the loop and at the resonator hot arm t i p was also recorded. A d i f f e r e n c e of about 7.5% between the computed and measured values was found. This was probably due to deviations of the resonator parameters from the computed ones and inaccurate p o s i t i o n i n g of the coupling loop. The discrepancy could e a s i l y be removed by r e p o s i t i o n i n g the coupling loop. The voltage phase was measured and plotted i n F i g . 69. - 116 " The following remarks give possible sources of discrepancies found during the t e s t s . i ) Since the c h a r a c t e r i s t i c impedance of the resonator was only calculated and not d i r e c t l y measured, an error up to 10% has to be considered. A l l computed quantities which were derived from Z q may be i n error. i i ) The 90 deg bend of the transmission l i n e was not taken into account, i i i ) The components were not selected with s p e c i a l precautions. The components were temperature dependent and some of the r e s i s t o r s were inductive. Also a l l leads connecting capacitors contributed i n d u c t i v e l y . i v ) A l l computing was based on the t h e o r e t i c a l values of q u a l i t y factors and C.],j.p. The e f f e c t of the Q on the transmission l i n e parameters can be found i n Section II.8. - 117 -CHAPTER IV. CENTRE REGION CYCLOTRON 1. LOW POWER LEVEL 1.1 Quality Factors As the actual q u a l i t y factors of the CR resonator are much higher than those measured on the h a l f - s c a l e resonators and because of frequent mechanical v i b r a t ions of the resonator panels (Q — 400. f = 4 Hz) the determination of mech the q u a l i t y factors by measuring a decay time constant of the resonator f i e l d s was preferred. The measured and computed q u a l i t y factors are compared i n Table XVII. TABLE XVII Measured and computed q u a l i t y factors of the CR resonator Computed Measured one Dee 7100 6300 Q 3 one Dee 12400 9000 two Dees 7100 6250 Q 3 two Dees 12400 6200 The q u a l i t y factors 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. A measurement with two Dees was done when they were i n s t a l l e d i n the vacuum tank (see Section IV.1.8). To achieve a high q u a l i t y f a c t o r the resonator segments had to be aligned very accurately and any leakage of the RF energy had to be avoided. Perfect e l e c t r i c a l contacts were necessary at the root of the resonator, both between the root plunger and the arms of one resonator segment, between the i n d i v i d u a l root plungers, and between the t i p s of the resonator segments. Moreover, a - 118 -good contact was desired between the root plungers and the f l u x guide shorts. When measuring the q u a l i t y f a c t o r of a one Dee resonator a good e l e c t r i c a l contact had to be provided between the grounded plate (simulating the v i r t u a l nodal plane) and the ground arm t i p s . It was observed that almost no energy was leaking out of the Dee when i t was accurately aligned and when good contacts were ensured. I f any leakage was l a t e r present i t was always due to i n s u f f i c i e n t contacts at the root rather than to the gaps between the resonator panels. Neither the coarse nor the fin e tuning of the Dee caused measurable changes i n the Q at the fundamental frequency. The coupling between the resonator and the o s c i l l a t o r had to be kept at an absolute minimum i n order to obtain correct measurements. A loose capacitive coupling placed at the main coupling loop opening was employed. 1.2 Resonator Frequencies The resonant frequencies of the assembled and thoroughly l e v e l l e d Dee were found to be 23.10 MHz and 69.29 MHz. To minimize the e f f e c t of the measuring devices on the frequency, a very loose capacitive coupling both between the o s c i l l a t o r and the resonator and between the frequency counter (eventually the scope) and the resonator was used. The o v e r a l l frequency s t a b i l i t y of the present resonator was about 1/10\ Due to inherent q u a l i t i e s of the hot arms, the temperature gradients i n the resonator panels caused mechanical d i s t o r t i o n s during the temperature transients. A uniform change i n temperature of the hot arm by AT = 80 °F (T = 70 °F, T 2 = 150 °F) resulted i n almost 1.2 cm d e f l e c t i o n of the hot arm t i p . The pressure of the c i r c u l a t i n g hot water was 35 p s i . By measuring the d e f l e c t i o n at several points i t was noted that the approximation of the deflected hot arm by a parabola was j u s t i f i e d . C a l c u l a t i o n shows that a d e f l e c t i o n of the hot arm t i p of about Ag = 1 cm (parabolic approximation) - 119 -causes the frequency to change by more than 0.8%. These large frequency s h i f t s proved to be troublesome during the high power t e s t s . This was e s p e c i a l l y true during the start-up procedure. In the f i r s t order approximation Ag/'g <* AT/T and so was A f / f . Beside the temperature another factor was also i n f l u e n c i n g the frequency, namely the pressure i n the cooling channels. By pressurazing the cooling system a r e l a t i v e l y large frequency change occurred. This was again due to small d e f l e c t i o n s of the hot arm caused by changes i n the cooling water flow pressure (expansion or contraction of the cooling channels). The measurement showed that a change i n pressure by Ap = +30 p s i resulted i n Af = -0.4%. Random fl u c t u a t i o n s of the water pressure and water v e l o c i t y excited mechanical v i b r a t i o n s of the hot arms at t h e i r natural mechanical frequency. The measured frequency s h i f t s can not be compared with t h e o r e t i c a l c a l c u l a t i o n because the combined e f f e c t of the weight of the panel and the pressure i n the channels resulted 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 with one Dee at medium power l e v e l showed that the frequency s t a b i l i t y at the fundamental frequency was 4 parts i n 10^ when no water was present i n the cooling channels. The s t a b i l i t y was an order of magnitude lower when c i r c u l a t i n g the water. I t was noted that i n s t a b i l i t i e s due to temperature changes disappeared as soon as equilibrium was achieved ( i . e . no temperature diffe r e n c e between the hot arm skin and the supporting I-beams). However i n s t a b i l i t i e s caused by c i r c u l a t i n g the cooling water continued. To minimize the i n s t a b i l i t i e s , spacers were l a t e r inserted between the upper and lower hot arms. 1.3 Fine Frequency Tuning The tuning of the system by means of tuning bellows mounted on root pieces was tested both at low and high power l e v e l s . This tuning i s yet to be - 120 -tested at high power l e v e l during actual fixed frequency operation. Quite a large spread i n the measured values was caused by an inaccurate reading of the a i r pressure i n the actuating bellows. The 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 pressure d i f f e r e n c e equals zero. Table XVIII presents the measured values of frequency changes when one Dee was mounted i n the frame. The pressure d i f f e r e n c e was substituted for changes i n the p o s i t i o n of tuning bellows i n order to compare the computed values with the measured ones (see F i g . 17). A discrepancy i s due to the assumption that the current density on the bellows surface was taken to be the same as that at the root. Measurements on h a l f - s c a l e resonators showed that the f i e l d density at the bellows surface was smaller than that at the root. Also, the f i e l d density depended on the p o s i t i o n of the bellows with respect to the root plane. The q u a l i t y factors were unaffected. 1.4 Coarse Frequency Tuning A provision was made to d e f l e c t the ground arm t i p s by means of voltage probe mechanisms. The r e s u l t s of resonator coarse tuning are presented i n Table XIX. The length of the deflected t i p was 35 i n . measured from the centre of the a c c e l e r a t i n g gap. The measured and computed values are shown i n F i g . 20. The t h i r d harmonic q u a l i t y f a c t o r varied during the t e s t . This was due to a poor contact between the ground arm t i p s and the grounded plate during the measurement. In addition some energy could have been leaking through the openings which ocurred next to the f l u x guides during the ground arm t i p d e f l e c t i o n . However, coarse frequency tuning obtained by d e f l e c t i n g the ground arm t i p s can not be considered as the f i n a l s o l u t i o n of the frequency tuning problem f o r a number of reasons. Besides the mechanical problems, the main problems with regard to RF operation concern 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 caused by tuning bellows p O 2 ( l b / i n . ) P. 1 2 ( l b / i n l ) Aft (mm) f o l (MHz) A f , o l (%) f o 3 (MHz) Af o 3 (%) f It 0 3 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 caused by ground arm t i p d e f l e c t i o n h a l f turns Ag (mm) f o l (MHz) Af oi (%> f 0 3 (MHz) Af 0 3 (%) f It 0 3 01 - 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 0 23.1000 0 69.2306 0 2.9969 + 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 - 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 (see Sections II.4.5 and IV.1.5). I t i s , however, strongly recommended that the d e f l e c t i n g mechanism be retained f o r adjusting the frequency r a t i o . This would be done only once p r i o r to high power operation. 1.5 RF Probes In order to measure the actual resonator voltage during cyclotron operation two methods w i l l be used. Capacitive pick-up probes w i l l inform us on the l e v e l of the resonator voltage, an accurate value of the resonator peak voltage w i l l be obtained by measuring the energy gain of p a r t i c l e s using beam probes. The fi x e d RF probes i n s t a l l e d on ei t h e r Dee were c a l i b r a t e d against a thermionic diode probe d i r e c t l y by touching 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 with about 500 V on resonator. Although the t i p loading e f f e c t resulted i n a frequency s h i f t of Af = 80 kHz at the nominal frequency of 23.1 MHz, no change i n the voltage amplitude could be detected. Changing the loading impedance connected to the probe output did not have measurable e f f e c t on the resonator 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 . The step-up r a t i o s were equal to 1500 at the middle range of the coarse frequency tuning adjustment and varied as the ground arm t i p s were deflected. No precise measurement of the resonator voltage amplitude can be c a r r i e d out unless the ground arm t i p s are at 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 voltage probes proved to be quite d i f f i c u l t when two Dees were assembled i n the vacuum tank. The loading e f f e c t and the presence of poor RF contacts seemed to be the main factors which influenced the c a l i b r a t i o n . I t was suspected that the v i r t u a l nodal plane i n the middle of the accelerating gap was being d i s t o r t e d . To make the voltage probes read the same voltage, the capacitive plates had to be repositioned so that a l l the - 123 -probes were at the same p o s i t i o n with respect to the ground arms. A good connection had to be secured f o r each probe. Due to mechanical v i b r a t i o n s of the hot arms the voltage measured during high power operation could be determined with an accuracy of only about 1%. The other method i s based on accurately measuring the energy of the p a r t i c l e s by means of movable scanning probe. A very accurate determination of the resonator voltage amplitude can then be made. During the RF tests t h i s probe could also be used to measure the voltage uniformity along the hot arm t i p s . This method has yet to be tested. 1.6 Coupling Loop Several tests r e l a t e d to the coupling loop self-inductance, the resonator t i p to loop voltage r a t i o and the vacuum seal were ca r r i e d out. The loop self-inductance was measured using Hewlett-Packard vector impedance meter with only the ground arms i n s t a l l e d i n the vacuum tank. Measuring the impedance of the loop at two d i f f e r e n t frequencies gave the values L^ = 0.221 uH and L^ = 0.212 uH. These were i n very good agreement with the computed value L = 0.224 uH. The resonator t i p to loop voltage r a t i o was measured for the nominal p o s i t i o n of the loop with the v e r t i c a l part of the transmission l i n e i n place. Several ways of e x c i t i n g the system have been t r i e d giving d i f f e r e n t r e s u l t s for d i f f e r e n t couplings between the Dee and the RF source. With the v e r t i c a l part of the transmission l i n e i n place i t was d i f f i c u l t to excite the system at the resonator resonant frequency. I t was shown i n Section I I . 10.3 that t h i s r a t i o varied with the d r i v i n g frequency. Although t h i s test f a i l e d we were quite c e r t a i n that the e x i s t i n g r a t i o was very close to the computed value because the p a r a l l e l impedance of a one Dee resonator measured with the vector impedance meter at the coupling point turned out to be very - 124 -close to the computed value. This test was done with one Dee only. Since the f i r s t attempts to make a good vacuum-tight t e f l o n s e a l did not succeed, the ceramic disc was temporarily replaced with a t e f l o n s e a l . Most RF high power tests were done with the t e f l o n seal which at some times presented many d i f f i c u l t i e s . The RF losses and stresses o r i g i n a t i n g from them seemed to present no d i f f i c u l t y . However, leaks developed several times and were probably due to mechanical forces at the coupling loop assembly. 1.7 Transmission Line and Resonances of the System Some problems were experienced i n the beginning due to the lack of good measuring equipment. The i n i t i a l tuning of the l i n e i n order to a t t a i n a desired input impedance was done using an oscilloscope and the Hewlett-Packard o s c i l l a t o r . The transmission l i n e with a dummy load at the end could e a s i l y be tuned f o r the proper input impedance, because i n d i v i d u a l resonant frequencies of the system were separated by more than a few MHz. Two Jennings capacitors were inserted i n the l i n e as indicated by the computer program RESLINE. The t h i r d capacitor, CP(6), was i n i t i a l l y i n s t a l l e d but l a t e r removed. I t was found that the plate-to-ground capacity together with the stray capacity i n the amplifier cabinet was of such a value that the t h i r d capacitor had almost no e f f e c t on tuning the l i n e . The capacitor settings of the Jennings capacitors which res u l t e d i n the required input impedance seemed to be i n disagreement with the computed values. This was p a r t i c u l a r l y true f o r the capacitor i n the middle of the l i n e . The charts supplied with the Jennings capacitors were obtained from tests at a low frequency. The measurement done at TRIUMF showed that at high frequencies the capacitors represented a capacitance i n s e r i e s with an inductance whose magnitude depended on the p a r t i c u l a r arrangement and RF contacts during the measurement. Our conclusion was that at high frequencies the capacitor values - 125 -TABLE XX Measured and computed resonances f or a one Dee-resonant l i n e system f computed (MHz) Type f measured (MHz) Type se r i e s or p a r a l l e l 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 68.000 S-P 75.739 S 88.340 P 89.884 S were a l i t t l e higher than those indicated on the charts. The transmission l i n e tuning became more d i f f i c u l t when a one Dee resonator replaced the dummy load. However, most troubles disappeared when the vector impedance meter was employed to measure the input impedance of the l i n e , and the accuracy of the measured quantities s i g n i f i c a n t l y increased. In p a r t i c u l a r , the resonant frequencies of the system could be determined, see Table XX. It was also proved that any desired value of input impedance (at the fundamental frequency) could be attained by a s u i t a b l e combination of the two - 126 -TABLE XXI Measured and computed resonances f o r a two Dee-resonant l i n e system f computed (MHz) Type f measured (MHz) 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 • • -P capacitors. I t was noted that some resonances of the system coincided with resonances of the power amplifier elements, which of course, caused p a r a s i t i c o s c i l l a t i o n s at those frequencies. The o r i g i n a l l y used narrow band am p l i f i e r could not cope with problems a r i s i n g from an unstable resonant load at the end of l i n e . The above mentioned tests were l a t e r repeated when a new tetrode amp l i f i e r replaced the triode narrow band a m p l i f i e r . Similar r e s u l t s were found. The res-onances of the two Dee resonator-transmission l i n e system were measured i n the range between 20 MHz and 25 MHz. Both the computed and measured values are presented i n Table XXI. The c a l c u l a t i o n of the resonances was done with the resonator frequency equal to 22.66 MHz. The current and voltage d i s t r i b u t i o n s i n the section of the l i n e between the resonator and the middle capacitor CP(3) turned out to be very close to computed values. The voltage node p o s i t i o n was found to be within 3 cm of the computed value. However, the conditions i n the section of the l i n e between the middle capacitor and the tetrode 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 . This was - 127 -accounted for by a very high capacity (plate-to-ground i n p a r a l l e l with a stray capacity) which was connected at the input of the l i n e . Due to the excessive amount of stray capacity capacitors CP(2) and, i n p a r t i c u l a r , CP(3), had to be readjusted to give a correct input impedance. 1.8 Comparison of Results from Lumped and Di 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 the input impedance of the system vs frequency, resonator voltage response vs frequency, beam loading etc., could only be carr i e d out i f our r e a l system with d i s t r i b u t e d parameters was replaced with lumped constants. The lumped parameters were computed at the resonant frequency of the resonator and i t was, therefore, necessary to determine the frequency range where they s t i l l represented our d i s t r i b u t e d system. The easiest check of v a l i d i t y of lumped constants was to d i r e c t l y measure the resonator voltage and input impedance as functions of frequency and compare them with r e s u l t s of computer code MATCH and with r e s u l t s obtained from a lumped parameter representation of our resonator-transmission 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 with one Dee resonator and two Dee resonator connected at the end of resonant l i n e . The input impedance was measured using the Hewlett-Packard vector impedance meter. The resonator voltage was d i r e c t l y measured on the Tectronix scope with a s i g n a l taken from the voltage probe at the resonator t i p . When measuring the voltage dependence a r e s i s t o r simulating the operating resistance of the amplifier was connected i n ser i e s with the o s c i l l a t o r . The input impedance was matched to the operating resistance at the resonant frequency of the resonator. Figs. 39, 40 show the re s u l t s from the tests when two Dees were loading the transmission l i n e . Comparison with the computed values p l o t t e d i n Figs. 39, 40 also revealed that - 128 -the representation of the system by three 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 being equal to 5 kHz. This was mainly due to a representation of a resonant l i n e by a sing l e lumped parameter resonant c i r c u i t . In t h i s range the measured quantities roughly followed the computed ones. From the curves i n Figs. 39, 40 one can also estimate the q u a l i t y factor of the whole system considered as a s i n g l e p a r a l l e l resonant c i r c u i t . The q u a l i t y f a c t o r of the resonator i s approximately equal to the q u a l i t y f a c t o r of the whole system (no impedance connected at the l i n e input) or twice as high 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 simulating the tube resistance i s connected at the input of l i n e . For a two Dee resonator the voltage curve, F i g . 40, y i e l d s Q = 3250, i . e . Q = 6500. The impedance curve, Li F i g . 39, gives Q = 6050. - 129 -2. HIGH POWER LEVEL 2.1 Sparking and Multipactoring High power tests started with a Dee mounted i n an a u x i l i a r y frame. The Dee was energized to a l e v e l of 40 to 50 kV i n a i r , where sparkovers at the resonator t i p s prevented the attainment of higher voltages. At the 50 kV l e v e l sparks occurred at the resonator t i p s i n the upper segments of the resonator, mainly between the hot arm-ground arm t i p s and across the ground arm-ground arm flu x guide gap. No sparks were observed i n the lower resonator segments or at the voltage probes. An improvement was achieved when the resonator was enclosed i n p l a s t i c sheets and f i l l e d up with an a i r - f r e o n mixture. The resonator voltage could be increased to 72 kV when discharges occurred again. Power loss i n the resonator was measured c a l o r i m e t r i c a l l y using two thermistors attached to the i n l e t and outlet cooling channels. The r e s u l t was within 5% of the computed value f o r 72 kV voltage l e v e l i n resonator. Resonator operation under high vacuum brought along the problems associated with multipactoring. With the pressure of the order of magnitude -5 10 Torr at the s t a r t of the tests and resonator panels not given any s p e c i a l cleaning, the resonator voltage could not be made to exceed a l e v e l of 170 V for a long period. This lower threshold multipactoring voltage corresponded very c l o s e l y to the calculated value of 180 V (see Section II.3.3). At th i s time a blue glow discharge was observed at the resonator t i p s . No d e t a i l e d analysis of the voltage wave form during multipactoring was done. The am p l i f i e r was then pulsed but i t was not u n t i l the pressure was improved to -6 10 Torr that we could break through multipactoring and the resonator was operated at high voltage l e v e l . For the next set of measurement the resonator panels were cleaned 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 at about 200 °F. - 130 -The pressure of the order of magnitude 10 6 was then e a s i l y obtained. Fewer problems due to multipactoring were expected from the t h e o r e t i c a l c a l c u l a t i o n s of the voltage r i s e time of a two Dee resonator and from the precautions taken with regard to the cleaning of the panels. Although i t seemed that m u l t i -pactoring occurred at low voltages i t could not be c l e a r l y i d e n t i f i e d . With a good broadband tetrode amplifier on hand and the a b i l i t y to put 140 kW into resonator ( i n pulse) i t was always possible to get to higher voltages within a few seconds. To check the l e v e l of the breakdown voltage i n the CRM, the accelerating gap was reduced to 2 inches from the nominal 6 inches. At the 120 kV l e v e l , (240 kV accelerating voltage), no sparks occurred, provided that the vacuum - 6 was better than 5 x 10 Torr. Sparks were usually experienced i n the beginning of high power t e s t s . A f t e r a period of conditioning i t was no problem to hold 100 kV on resonator f o r several hours. However, at times sparks occurred owing to f a i l u r e s i n contacts at the root. Eventually, the poor contacts at the root were responsible for RF leakage which gave r i s e to RF discharges r e s u l t i n g i n a blue glow everywhere ins i d e the vacuum tank. With a good vacuum seal neither sparking nor multipactoring occurred around the coupling loop. In one instance ion sputtering was i n i t i a t e d by a small leak i n the vacuum seal r e s u l t i n g i n large metal deposits on the t e f l o n d i s c and on the inner 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 softened and was bent inwards. The current carrying fingers underneath the i n s u l a t o r l o s t contacts, small sparks and burning of fingers followed and t h i s l e d to the destruction of the s e a l . The e f f e c t of the magnetic f i e l d of the magnet on the RF operation was also investigated and found undetectable. - 131 -2.2 RF Contacts To obtain good RF contacts rectangular springs (Melrose, #60189), fingerstocks (Eimac, #CF-800 and CF-900) and fuzz-buttons (Tecknit, #5024/0431/750) were employed (see F i g . 71). At high power l e v e l they had to withstand severe mechanical and e l e c t r i c a l stresses. The damage to both the springs and fingerstocks was usually due to sparking which was caused by imperfect RF contact. A very poor RF contact existed o r i g i n a l l y between the f l u x guide end sections and the root pieces. Both t h i s and the mechanical misalignments between the flux guides and the resonator panels influenced the e l e c t r i c a l properties of the resonator. At high power l e v e l the amount of energy leaking out of the resonator due to poor contacts at root was quite s i g n i f i c a n t and as a r e s u l t discharges occurred behind the resonator. The contacts were improved when new f l u x guides were i n s t a l l e d . I t should be pointed out that as long as the good e l e c t r i c a l contact was maintained both springs and fingerstocks performed s a t i s f a c t o r i l y . Fuzz-buttons and l a t e r s l i d i n g springs provided a good RF contact between the hot arm t i p s and between the hot arm t i p and f l u x guide. No damage due to sparks at the t i p s was observed. To enable v e r t i c a l motion of the ground arm t i p s during the coarse frequency tuning, S-shaped pieces of 0.005 i n . thick brass sheet were used to provide a contact between the beam probe housing and the ground arms. Owing to mechanical stresses exerted upon the coupling loop the f i n g e r contacts between the loop and the ground arm were damaged several times and had to be replaced. 2.3 Voltage and Frequency S t a b i l i t y In the beginning i t was found d i f f i c u l t to feed RF power into the - 132 -resonator for two reasons: multipactoring and temperature transi e n t s . With no spacers between the hot arms the frequency of the resonator varied by f a r more than 1% when the multipactoring range was passed and power was fed to the resonator. Since the o r i g i n a l triode a m p l i f i e r was narrowband, i t was always necessary to tune and match the a m p l i f i e r to the ion loading conditions i n order to punch through multipactoring. Once the multipactoring load disappeared a d i f f e r e n t impedance was presented to the a m p l i f i e r . Consequently, the a m p l i f i e r had to be retuned and rematched. More power was then fed to the resonator and more heat was, therefore, developed. This resulted i n large frequency s h i f t s due to thermal d e f l e c t i o n s of the resonator panels. A new retuning was required and the cycle was repeated. The only remedy was to i n s e r t spacers i n the beam gap and thus lock the hot arms i n t h e i r p o s i t i o n . This improved the s t a r t up and enabled the attainment of a higher voltage l e v e l , however, the i n s t a b i l i t i e s (Af/f = ±0.01%) due to water pressure f l u c t u a t i o n s (±1/2 psi) remained. No d e t a i l e d frequency and voltage s t a b i l i t y analysis was made at this stage because i t was never possible to run the resonator under r e l a t i v e l y steady conditions f o r a long time. New tests with an improved cooling system and with the tetrode a m p l i f i e r broadband ne u t r a l i z e d proved to be more s u c c e s s f u l l . Since the resonator panels were cleaned i n a 10% water s o l u t i o n of #33 Oakite i n a wooden u l t r a s o n i c cleaning tank before i n s t a l l i n g them into the vacuum tank there —6 were no problems i n achieving good vacuum of the order of magnitude 10 Torr. This f a c t o r and the f a c t that a 3 inch gap was no longer present (a two Dee resonator) influenced the s t a r t up procedure. A pulsing device with a v a r i a b l e amplitude and width of a pulse was used to break through multipactoring. No d i f f i c u l t i e s were experienced t h i s time. Once multipactoring was passed and the resonator voltage reached about 20 kV a switch to a s e l f - o s c i l l a t o r y mode - 133 -occurred. Now the resonator voltage detector output was compared with a reference voltage and the error s i g n a l was amplified to drive e i t h e r the input modulator or the screen modulator. The input modulator was employed when the voltage l e v e l i n the resonator was to be increased from about 20 kV to i t s f i n a l value. The screen modulator, however, was used to maintain a steady RF voltage amplitude at a given l e v e l . At approximately 100 kV l e v e l the frequency s t a b i l i t y was ±0.01% and the voltage s t a b i l i t y was ±0.01%. It should be noted that since the automatic frequency c o n t r o l was not yet i n operation the power ampl i f i e r had to be driven by a s i g n a l derived from the resonator a l l the time. The RF system was thus s e l f - o s c i l l a t o r y at the resonator frequency. This allowed considerable resonator d r i f t during warm-up. At high voltage l e v e l when the resonator frequency has s t a b i l i z e d the ampli f i e r w i l l be driven by a synthesizer. 2.4 Coupling Loop and Transmission Line It was found that the coupling loop could be adequately cooled by the addition of a cooling channel around the h o r i z o n t a l edge of the loop. However, the v e r t i c a l part of the transmission l i n e next to the coupling loop was warm due to the conduction of the heat developed i n the v e r t i c a l section of the coupling loop. The losses i n the transmission l i n e section near the loop due to the input current were small enough i n the CR cyclotron but they w i l l increase by a factor of 100 i n the main cyclotron i f the same dimensions are used (see Section II.7.2). Due to a lower q u a l i t y factor of the resonator and because the outer conductor of the l i n e was made of aluminum, the power loss i n the part of the l i n e between the loop and the middle capacitor might have been higher by a fa c t o r of 1.5 than the computed value of 500 W. Also because the capacitance - 134 -around the tube was very high, the capacitors, e s p e c i a l l y the middle 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 standing wave r a t i o i n the l i n e near the tube but increasing the standing wave r a t i o i n the f i r s t and second sections of the l i n e . Fresh a i r was l a t e r blown i n the transmission l i n e i n order to reduce heating of the l i n e i n the f i r s t and second sections. Later when the l i n e was taken apart, large deposits of oxide were found near the voltage node. Depending on the density of the co l o r i n g , the node was determined to be within 3 cm of the computed value. Several sparks occurred i n the transmission l i n e near the middle capacitor. These sparks were always due to moisture and water drops o r i g i n a t i n g from the cooling l i n e s f o r the middle capacitor. Otherwise no problems were experienced with the transmission l i n e during high power te s t s . 2.5 RF Power Amplifier The o r i g i n a l design proposed the use of a ML-7560 triode operating i n s e l f -biased, grounded cathode configuration. This a m p l i f i e r u t i l i z i n g c o i l n e u t r a l i z a t i o n did not perform s a t i s f a c t o r i l y when loaded by an unstable res-onant load. Since t h i s form of n e u t r a l i z a t i o n was e f f e c t i v e at only one frequency, the resonant n e u t r a l i z i n g c i r c u i t had to be readjusted each time the resonator frequency changed. In addition, there were problems with p a r a s i t i c o s c i l l a t i o n s . Consequently, a new design was required. The present CRM RF amp l i f i e r , designed and b u i l t by Continental E l e c t r o n i c s , 1 8 employes the Eimac 4CW250,000 tetrode i n a grounded cathode configuration. The single-ended 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 of p a r a s i t i c o s c i l l a t i o n s f o r a wide band of frequencies. 1 8 The RF am p l i f i e r f o r the main cyclotron RF system consists of four power amp l i f i e r s , each using a p a i r of 4CW250,000 tetrodes i n a push-pull grounded gr i d c i r c u i t . - 135 -CHAPTER V. SUMMARY AND CONCLUSIONS The experimental tests c a r r i e d out to date have shown that the resonator c h a r a c t e r i s t i c s are very close to the predicted ones. The measured resonator parameters such as the power l o s s , the resonator lumped capacity and the loop self-inductance were within 5% of the computed values. Good q u a l i t y factors were measured on a one Dee resonator (CRM)rQ^ = 6300 (7100), Q 3 = 9000 (12400). When two Dees were assembled, the fundamental Q remained the same, but the t h i r d harmonic Q dropped to about 6200. The q u a l i t y f a c t o r s , resonant frequencies and the power losses were found to be s i g n i f i c a n t l y influenced by alignment errors and by poor contacts between the resonator segments. Several methods of resonator frequency tuning were v e r i f i e d on modelled res-onators at h a l f - s c a l e . Only two of them, ground arm t i p d e f l e c t i o n and tuning bellows at the root, s a t i s f i e d the c r i t e r i a placed on the voltage v a r i a t i o n along the accelerating gap (<.5% i n the c e n t r a l region) and q u a l i t y f a c t o r v a r i a t i o n (<3%). Although a tuning method by means of a ground arm t i p d e f l e c t i o n would normally be preferred because of larger frequency range, for engineering reasons and because of possible problems with the RF f l a t - t o p operation, tuning using the bellows at the root was adopted. A frequency change of A f ^ +.02% f o r a bellows motion of Aft - -.25 i n . was measured on prototype resonators (CRM). High power tests have proved that the tuning bellows system i s designed mechanically so as to withstand high currents and the r e s u l t i n g heating. Automatic frequency co n t r o l ( A f ^ ±.02%) w i l l then be accomplished by including the bellows system i n the RF feedback path. This i s yet to be tested. The voltage s t a b i l i t y achieved when running the RF system i n s e l f - o s c i l l a t o r y mode at the 100 kV l e v e l was ±.01%, the frequency s t a b i l i t y was also ±.01%. The desired tolerances are summarized i n Table I. The i n s t a b i l i t i e s were mainly due to d e f l e c t i o n s of the resonator hot arms caused by temperature changes and by cooling water pressure v a r i a t i o n s . A new design of the resonator panels - f l o a t i n g skin design - w i l l be used i n the main cyclotron (the d e f l e c t i o n due to temperature changes reduced - 136 -by a factor of 10). Although the cooling system has already been improved, a few a d d i t i o n a l improvements may be required i n order to reduce the present water pressure f l u c t u a t i o n s (±.5 p s i ) . The design of a coarse frequency change of 3% ( s h i f t of a root plane) was abandoned due to engineering problems and a very t i g h t time schedule. Tuning diaphragms, desired during RF f l a t - t o p operation i n fo3 order to a t t a i n an exact frequency r a t i o =3, were su c c e s s f u l l y tested at low oi power l e v e l on a resonator modelled at f u l l - s c a l e . No f i n a l decision regarding the use of the diaphragms i n eit h e r the CR cyclotron or main cyclotron has yet been made. Multipactoring presented many problems during high power operation with one Dee only. These problems seemed to disappear when a two Dee resonator was used. However, such precautions as cleanliness of the resonator panels, pulsing of the RF a m p l i f i e r and a good pressure of the order 10 Torr were mandatory i f m u l t i -pactoring was to be overcome. Sparks usually occurred during a period of conditioning. A f t e r conditioning i t was possible to hold a stable Dee-to-Dee g voltage of 200 kV for several hours, provided the pressure was less than 5x10 Torr. A si n g l e coupling loop excited s u c c e s s f u l l y a two Dee resonator. The ceramic vacuum feedthrough f a i l e d due to mechanical stresses and had to be temporarily replaced by a t e f l o n s e a l . Design of a new ceramic feedthrough i s i n progress. A resonant transmission l i n e performed s a t i s f a c t o r i l y . A desired impedance match could e a s i l y be attained by making use of two capacitors connected along the l i n e . However, since the resonator frequency was not stable (±.01%), one could only say that an impedance match was obtained at a frequency very close to the resonator frequency (within 5 kHz). Resonant operation at a frequency d i f f e r e n t from the resonator frequency (Af > 1 kHz) would lead to higher losses i n the coupling loop assembly and, therefore, 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 of the resonator with the centre post and cut-out included are presently being v e r i f i e d (CR cyclotron). - 137 -REFERENCES 1. E.W. Vogt and J . J . Burgerjon, e d i t o r s , "TRIUMF Proposal and Cost Estimate" (1966) 2. E.G. Auld et a l . , "Design of the 4000 Ton Magnet f or the TRIUMF Cyclotron" i n Proceedings, International Cyclotron Conference, 5th, Oxford, 1969 (Butterworths, London) 3. L. Root and E.W. Blackmore, private communication (1970) 4. L.P. Robertson et a l . , "Extraction of Mult i p l e Beams of Various Energies from the TRIUMF Negative Ion Isochronous Cyclotron" i n Proceedings, International Cyclotron Conference, 5th, Oxford, 1969 (Butterworths, London,1971) 5. J.R. Richardson and M.K. Craddock, "Beam Quality and Expected Energy Resolution f o r the TRIUMF Cyclotron" i n Proceedings, International Cyclotron Conference, 5th, Oxford, 1969 (Butterworths, London, 1971) 6. K.L. Erdman et a l . , "A 'Square Wave' RF System Design f o r the TRIUMF Cyclotron" i n Proceedings, International Cyclotron Conference, 5th, Oxford, 1969 (Butterworths, London, 1971) 7. G.M. Stinson et a l . , " E l e c t r i c D i s s o c i a t i o n of H Ions by Magnetic F i e l d s " , TRI-69-1 (1969) 8. E.W. Blackmore and G. Dutto, pr i v a t e communication (1971) 9. J.R. Reitz and F.J. M i l f o r d , Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1967) 10. 0. Dambach, pri v a t e communication (1971) 11. R. Gummer, private communication (1971) 12. J.R. Richardson, pr i v a t e communication (1969) 13. J.R. Richardson, private 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 Sparking Designed to Include both RF and DC", Report U.C.R.L.-2321 (1953) 15. B.H. Smith, "Radio-frequency System of the Berkeley 88 Inch Cyclotron", Nucl. I n s t r . & Meth. 18 (1962) 16. J.C. Sl a t e r , Microwave Transmission (McGraw-Hill, New York, 1942) 17. Reference Data f o r Radio Engineers, 5th e d i t i o n (Howard W. Sams, Indianopolis, 1969) 18. Proposal f o r a 150 kW Amplifier, a 1.5 MW RF Amplifier and a 2.4 MW Power Supply, Continental E l e c t r o n i c s Manufacturing Co., Dallas, Texas (1971) iOOkv/ Particle Frequency- 4.62 R.F Frequency•• 23 R.F SKIN LOSSES TOTAL R.F POWER R.F COUPLING LOOPS FLUX COUPLINGS + IOOkV + IQOkV 200kV 4>-O F i g . 3 Resonator system synthesizer RF sample remote auto/synthl l o c a l _ auto/synth manual RF l e v e l — input selec. phase det. safety c o n t r o l ' input -?=1 mod. ohase s h i f t fundamental IF a m p l i f i e r 3rd *** harmonic I trans, l i n e screen -s> supply screen operating point 3rd harmonic 3rd harm, input '"amplifier trans. l i n e fundi > f r e q . meter esonator root current loop current J phase det. tuning plunger d r i v e r frequency ranslato 3rd harmonic r e l . amplitude: kphase mate set 3rd harm.-amplitude mod. input loop loop amp. amp. ... pnase e r r o r to t r i p l e r i-O screen operating point set 3rd harmoni phase F i g . 5 Radio-frequency system 1600 - 143 -RF fundamental voltage amplitude V (kV) 80 90 100 110 120 S 1400 u CO 3 1200 IOOO L 800 L 600 L 400 1400 to CO o u o PH 1200 L IOOO r — 800 L 600 c h a r a c t e r i s t i c impedance of segment Z (ft) Power loss i n two Dees a) as a function of s p e c i f i c conductivity and RF fundamental voltage amplitude b) as a function of c h a r a c t e r i s t i c impedance and resonator fundamental frequency o U4 ^ 4 0 0 0 2000 L 2000 4 5 6 7 s p e c i f i c conductivity a (Mhos/m) 8 F i g . 7 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 of resonator segment ZQ b) vs s p e c i f i c conductivity o and resonator fundamental frequency f F i g . 8 Resonator hot arm d e f l e c t i o n a) resonator made up of n sections b) l i n e a r approximation c) parabolic approximation t i p d e f l e c t i o n Ag (mm) F i g . 9 Percentage frequency change vs hot arm d e f l e c t i o n - 147 -M d) =F C R L -o- - A A A A / V -R i 6 F i g . 10 Lumped parameter representation of the system a) system consists of a s i n g l e resonator b) system consists of a one Dee resonator and a resonant line, c) system consists of a two Dee resonator and a resonant l i n e d) equivalent representation f o r a) 150 15 > ^4 100 0) -a 3 a; 60 CO o CO c o CO ai 10 f i r s t harmonic, f = 22.66 MHz, two Dee resonator and resonant l i n e t h i r d harmonic, f = 67.98 MHz. two Dee °3 resonator and resonant l i n e f =22.66 MHz, one Dee oi f i r s t harmonic resonator f i r s t harmonic, f =22.66 MHz, one Dee resonator and resonant l i n e 0 50 100 t (usee) 150 2 0 0 F i g . 11 Resonator voltage amplitude r i s e during transient - 1 4 9 -0 3 one Dee resonator r two Dee resonator and resonant l i n e , Main cyclotron two Dee resonator and resonant l i n e , CR cyclotron — one Dee resonator and resonant l i n e , CR cyclotron two Dee resonator and resonant l i n e , CR 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 0 .0 25 5 0 t (usee) F i g . 1 2 I n i t i a l r i s e of resonator voltage amplitude 4 hot arm length I (cm) F i g . 13 Resonator resonant frequency vs hot arm length F i g . 14 Resonator resonant frequency vs t i p loading capacity - 153 -b) —4 1 i i > s o s. 4- e a /o gap Ag (mm) Fi g . 16 Percentage frequency change vs p o s i t i o n of cap a c i t i v e plates a) the fundamental b) the t h i r d harmonic a) - 154 -parabolic approximation measured values l i n e a r approximation f , = 23.10 MHz ol - 6 -4 -2 0 t i p d e f l e c t i o n Ag (mm) 1 1 1 _ parabolic approximation measured values l i n e a r approximation f , = 69.27 MHz o3 6 _4 t i p d e f l e c t i o n Ag (mm) 6 F i g . 17 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 a) - 155 6^  CD 60 is Xi o CJ C cu 3 cr cu u - I - 2 - 3 - 4 - 5 b) IOOOO 8G00 u o o ca w 6000 u rH H CO cr 4000 2000 computed values, f ~X measured values, f o l 'measured values, f 0 3 • computed values, f 0 3 139.52 MHz 139.52 MHz 6 8 10 12 t i p d e f l e c t i o n Ag (mm) T —o o measured values, f i r s t harmonic measured values, t h i r d harmonic X X -o-_X_ _o_ X -x--o-8 10 t i p d e f l e c t i o n Ag (mm) 14 -X. 14 16 x -o 16 F i g . 18 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 ground arm t i p d e f l e c t i o n ( h a l f - s c a l e resonator). - 156 -F i g . 19 Schematic of the tuning bellows - 157 -Ll (mm) F i g . 20 Percentage frequency change vs p o s i t i o n of tuning bellows (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 perturbation theory by S l a t e r . F i g . 21 Percentage frequency change vs p o s i t i o n of diaphragm i n the resonator c a l c u l a t e d using perturbation 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 root; the diaphragm i s i n i t s v e r t i c a l position) p o s i t i o n of diaphragm (cm) F i 8 ' 2 2 F i g . 23 Extreme end se c t i o n a) top view b) section with modified hot 24 -Central resonator segments a) top view b) side view c) g r i d of points f or voltage measurements 25 Beam-RF f i e l d i n t e r a c t i o n a) resonator fed by an RF generator and a beam current generator b) amplitude of a beam pulse c)representation of a resonator with a beam load d) beam induced voltage and t o t a l voltage on resonator F i g . 26 Power delivered to the beam a) and frequency change b) vs i n j e c t i o n phase (the RF fundamental, I = 100 uA, E = 500 MeV) f = 22.66 MHz f = 67.98 MHz 03 q + p = 80 deg one Dee resonator b) 500 N M < CJ 60 rt •d u o c o* CJ u 250 -250 L_ I L 90 - 60 - 30 0 -500 JL _L 30 60 90 90 -60 -30 0 30 60 90 pulse position (degrees) pulse position (degrees) Fig. 27 Power delivered to the beam a) and frequency change b) vs injection phase (the third harmonic, I = 100 uA, E = 500 MeV) F i g . 28 Power delivered to the beam a) and frequency change b) vs i n j e c t i o n phase (the RF fundamental, I = 750 uA, E = 400 MeV) f , = 22.66 MHz ol a) 10 * H 0 CJ u - 1 0 - 9 0 - 6 0 - 3 0 f - 67.98 MHz 03 q + p = 80 deg one Dee resonator b) 2.5 N <4-( < QJ 60 C ca x u o c cu 3 cr CD w C H 0.0 2.5 pulse p o s i t i o n (degrees) - 9 0 - 6 0 - 3 0 0 30 60 pulse p o s i t i o n (degrees) 90 F i g . 29 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 B = 750 uA, E = 400 MeV) f = 22.66 MHz ol f = 67.98 MHz 0 3 a) 30.0 CD 5-1 f-27.5 25.0 one Dee resonator b) 0.0 P-I CU u 25 _ - 5.0 • 1 1 1 ' ^ ~ — centre of pulse set at 0 deg of RF phase centre of pulse set at -20 deg of RF phase 1 1 l • 20 40 60 80 beam phase width (degrees) 100 20 40 60 beam phase width (degrees) 80 100 F i g . 30 Power delivered to the beam vs beam phase width a) the RF fundamental b) the t h i r d harmonic (I = 100 uA, E = 500 MeV) I 04 o frequency change Af (Hz) ro i ro o OQ t—' CT I T ) w l-t ft) rt ,n CD tt> 3 rt O H. O a. 3" cu 3* 3 So TO t-i fD 3o 3 H-O cn to CO 3 XS cu CO tt> S3 H> O o •c > o o s It) < 3* CD H i 3 3 Cu tt) 3 rt Cu ft) cu 3 XS 3" Cu CO ft) Cu rt . 3* Cu ft) TO i-l ft) ft) CD o —I 00 KJ ON cy. frequency change Af (Hz) o O O O o o 3 ft> a to ft> n fD CO o 3 cu rt o 1-1 - 89T -F i g . 32 T o t a l RF fundamental voltage on resonator during the i n j e c t i o n of a beam (without the RF t h i r d harmonic) F i g . 33 T o t a l RF fundamental voltage on resonator during the i n j e c t i o n of a beam (with, the RF t h i r d harmonic) 12 ; E g - 500 MeV, I = 100 yA — E g = 400 MeV, I g = 750 yA V o l - 113 kV, V o 3 = 13 kV -p = -36 deg q = 36 deg 0 100 200 t - t . (ysec) 300 4 0 0 F i g . 34 T o t a l RF t h i r d harmonic voltage on resonator during the i n j e c t i o n of a beam (with the RF t h i r d harmonic) - 172 -RF fundamental voltage wave \ flat-topped voltage wave A \ \ — — RF t h i r d harmonic voltage wave / / w - - - - -\ \ / 7 V V 211 \ \ / \ \ -p = - 52 deg, q =-28 deg -p = - 12 deg, q = 12 deg i_„ : o • II . 2 i r CO t 35 Components of resonator voltage during the course of ac c e l e r a t i o n a) RF voltage amplitudes due to external sources b) the f i r s t harmonic component of beam induced voltage (RF fundamental operation) p = -36 deg, q = 36 deg F i g . 36 Components of resonator voltage during the course of ac c e l e r a t i o n (RF f l a t - t o p operation) a) the f i r s t harmonic component of beam induced voltage b) the t h i r d harmonic component of beam induced voltage - 175 -a) B RL - A A / V v: v, B o-•o F i g . 38 Representation of coupling a) and lumped parameter representation of a Dee-resonant l i n e system b) measured values, f o l 22.937 MHz computed values (lumped parameter representation), f Q l = 22.660 MHz a) _ computed values (program MATCH), f 22.660 MHz 1000 eg CW O CU 750 500 _ cfl S3 250 - 5 . 0 -Z5 0 2.5 frequency change Af (kHz) 5.0 - 5 . 0 -2 .5 0 2.5 frequency change Af (kHz) 5-0 F i g . 39 Input impedance vs detuning from resonance a) magnitude of input impedance b) phase of input impedance - 177 -150 2.5 0.0 Af (kHz) 2-5 5-0 measured values - computed values (lumped parameter representation) F i g . 40 Resonator peak voltage vs detuning from resonance "A" DEE "B"DEE Tn lite l a . B DETAIL A T7 y L-L DETAIL B 3 Te '7 A T 5 E - E / .D T i L = length A = height C = thickness D = width DETAIL C T 6 T 4 T 2 z c 5 T I o H Co " C 6 < R T U B E F i g . 41 Representation of the system with d i s t r i b u t e d parameters (program RESLINE) - 179 -A - o -'TUBE nA/2 JOLL B • o , B' ± A o-XL+X TL B Xc V, A' B' Matching using a nA/2 l i n e a) representation of a Dee with 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 tQ c) impedance transformation using IT network F i g . 43 Representation of the system a) two Dees represented with lumped parameters and the transmission l i n e with d i s t r i b u t e d parameters (program MATCH) b) lumped parameter representation of the whole system - 181 -f = 23.1 MHz o - 1 0 0 10 d r i v i n g frequency change Af (kHz) F i g . 44 Phase, s h i f t between loop and resonator root currents vs d r i v i n g frequency - Z8T -- 183 -a) - 4 0 4 8 12 16 stub shorting plunger p o s i t i o n Ad (cm) b ) OB K 0.6 <] cu oo I 0.4 u c 0) S* Q2 M-l 0.0 f = 47.61 MHz oi a , - x/4 stub o Ad <* A/4 - I stub 1 4 8 12 16 stub shorting plunger p o s i t i o n Ad (cm) 46 Percentage frequency change vs stub shorting plunger 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 of 10 sections b) 4 stubs coupled c a p a c i t i v e l y to a two Dee resonator, each Dee made up of 10 sections F i g . 47 Voltage v a r i a t i o n along the accelerating gap vs stub shorting plunger 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 of 10 sections b) 4 stubs coupled c a p a c i t i v e l y to a two Dee resonator, each Dee made up of 10 sections T 1 1 r F i g . 48 Percentage frequency change and q u a l i t y factor v a r i a t i o n vs stub shorting plunger 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 of 5 sections) - 186 -F i g . 49 Voltage 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 of 5 sections) a) the fundamental b) the t h i r d harmonic a) - 187 -< cu CO c n) J2 o >. o c OJ 3 cr a) u 0.0 - 0 - 2 - 0 . 4 - 0 . 6 f = 46.337 MHz ol f „ = 139.608 MHz o3 -0 .8 0.0 b) 5000 T ± ± 0.5 1.0 1.5 2.0 p o s i t i o n of capacitors (cm) T T 2.5 4000 -© 3 2000 3 0* 1000 -• f i r s t harmonic Q S 3000 CJ ca -A t h i r d harmonic Q 0 0.0 0.5 1.0 1.5 p o s i t i o n of capacitors (cm) 2.0 2.5 F i g . 50 Frequency tuning by means of c y l i n d r i c a l capacitors (Dee made up of 10 sections) a) percentage frequency change and b) q u a l i t y factor v a r i a t i o n vs p o s i t i o n of c y l i n d r i c a l capacitors - 1 8 8 -F i g . 51 Voltage v a r i a t i o n along the hot arm t i p s vs p o s i t i o n of c y l i n d r i c a l capacitors a) 5000 4000 189 -u o 4-> o £ 3000 4-1 3 2000 1000 T : — i 1 — — f i r s t harmonic Q A A— t h i r d harmonic Q f = 45.2 MHz ol f = 137.4 MHz - o3 _L QO 0.2 0.4 0.6 gap Ag (cm) 0.8 b) > < 0) 60 ct) Xi o CU 60 « 4-1 rH O > 0 - 2 f. = 45.2 MHz < X-A--x * x x x -A A A A A A. ^ > A" -A -A~ - < - » — • o -"A 5~ A~ ± J_ 0.0 0.2 0.4 0.6 gap Ag (cm) 0.8 F i g . 52 Quality f a c t o r v a r i a t i o n a) and voltage v a r i a t i o n along the hot arm t i p s b) vs p o s i t i o n of c a p a c i t i v e tuning plates (Dee made up of 10 sections) a) b) p o s i t i o n of f i n s (degrees) p o s i t i o n of f i n s (degrees) F i g . 53 Frequency tuning by means of rot a t i n g f i n s (measurements done with a 3 s e c t i o n resonator with the f i n s inserted i n the upper and lower centre 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 vs p o s i t i o n of tuning f i n s F i g . 54 Frequency tuning by means of rot a t i n g f i n s (measurements done with a two se c t i o n resonator with 8 f i n s per 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 a ) b) 2.0 MH < 60 c (0 o >. U C 0) 3 cr CU M .0 0.5 0.0 Rl.5 -f = 47.44 MHz ol f , = 143.17 MHz o3 30 60 p o s i t i o n of loops (degrees) 90 10000 7500 u o •w o 4H 5000 •H 3 cr 2500 1 ' 1— _ t h i r d harmonic Q - A & f i r s t harmonic Q I 30 60 p o s i t i o n of loops (degrees) 90 F i g . 55 Frequency tuning by means of rot a t i n g loops (measurements done with a two s e c t i o n resonator with 8 loops per 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) T r i p l e r Power a m p l i f i e r O s c i l l a t o r D i g i t a l voltmeter T I I U Resonator 4 — n — V 4 -4 Harmonic analyzer T T " I I L J 4 " 4 TE JTA vector voltmeter Frequency counter C.R.T. b) F i g . 56 Resonator frequency tuning by means of diaphragms a) schematic of the experimental arrangement b) resonator with tuning diaphragms - 19,4 -a) b) <u 60 c CD X ! O o c 0) 3 a* <u M M O a cu 3 c r CD S-1 0.0 -0.5 -1.0 -1.5 -2 .0 0 3.02 3.00 2.98 -2.96 -f = 22.6508 MHz ol _J I I 30 60 90 p o s i t i o n of diaphragms (degrees) 30 60 90 p o s i t i o n of diaphragms (degrees) F i g . 57 Frequency tuning by means of tuning 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 of diaphragms; p o s i t i o n of diaphragms i n resonator at 108.5 i n . from root - 195 -a) b) < CU § o >> o c CU 3 cr cu u m ,-0.5 -1.0 15 -2 .d 0 3.02 3.00 2.94 0 •f . 21.65 i n . from root ol -f . 65.4 i n . from root o3 - A -_ L _ f ,65.4 i n . from root o i ' f ,21.65 i n . from root o3 30 60 90 p o s i t i o n of diaphragms (degrees) T T I "x x 65.4 i n . from root -e © — 21.65in. from root-f = 67.894 MHz 03 30 60 90 p o s i t i o n of diaphragms (degrees) Fig'. 58 Frequency tuning by means of tuning diaphragms a) percentage frequency change and b) t h i r d - t o - f i x 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 resonator at 65.4 i n . and 21.65 i n . from root - 19.6 -Fi g . 59 Resonator frequency tuning by means of a tuning stub a) experimental arrangement b) lumped constant representation F i g . 60 Percentage voltage 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 of the upper hot arm #3 i n an 18 . s e c t i o n one Dee resonator 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 resonator b) - 4 - 2 0 2 4 - 4 - 2 0 2 4 number of turns (0.05 in./turn) number of turns (0.05 in./turn) Fig.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 of the upper hot arm #3 i n an 18 section one Dee resonator 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 res F i g . 62 Percentage voltage change vs root plunger p o s i t i o n a) motion of root plungers i n section #7 i n an 18 s e c t i o n one Dee resonator b) motion of root plungers i n s e c t i o n #7A i n a 20 s e c t i o n two Dee resonator a) U-l < 01 00 o o c <D cr 0) u 0.2 0.1 0.0 0.1 -0.2 - 200 -r— f = 47.781 MHz o i - 4 -2 0 2 4 p o s i t i o n of root plungers Aft (cm) b) 0.2 -2 0 2 4 p o s i t i o n of root plungers Aft (cm) F i g . 63 Percentage frequency change vs root plunger p o s i t i o n a) motion of root plungers i n s e c t i o n #7 i n an 18 section one Dee resonator b) b) motion of root plungers i n section #7A i n a 20 section two Dee resonator a) 0.5 - 201 -I 0 I 2 p o s i t i o n of root plungers 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 0 I 2 p o s i t i o n of root plungers A£ (cm) Fig . 64 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 root plungers i n section #4 i n a 5 section one Dee resonator //// /// I'll //// In III (' III II IP ll II ill EQUIPOTENTIALs\\\ A U F i g . 65 Central region geometry a. 300 200 0) oo rt o > 100 -T 1 measured values computed values T capacitor CP(3) = 400 pF p o s i t i o n of CP(3): 342 cm capacitor CP(2) = 100 pF capacitor CP(6) = 87 pF POiNT OF REFERENCE 16 LOOP 24 26 28 number of s t a t i o n 29 30 31 32 33 TUBE Fig . 66 Voltage d i s t r i b u t ion along the transmission l i n e ; b a s i c set—up 300 I 1 -measured values •computed values T T 200 POINT OF REFERENCE OJ 4-1 100 capacitor CP(3) = 415 pF po s i t i o n of CP(3) » 342 cm capacitor CP(2) = 100 pF capacitor CP(6) = 70 pF = 2345 ft 32 33 LOOP number of s t a t i o n TUBE F i g . 67 Voltage d i s t r i b u t i o n along the transmission; l i n e ; magnitude of CP(3) changed to 415 pF 300 200 100 -9- -measured values ..computed values POINT OF REFERENCE capacitor CP(3) = 400 pF p o s i t i o n of CP(3) =327 cm capacitor CP(2) = 100 pF capacitor CP(6) = 74 pF 0 16 18 LOOP 24 26 28 number of s t a t i o n 29 30 31 32 33 TUBE Fi g . 68 Voltage d i s t r i b u t i o n along the transmission l i n e ; p o s i t i o n of CP(3) changed to 327 cm 360 270 0) oo rt 4-1 r H o 180 90 L-I I--measured values ^computed values ^ P O I N T O F R E F E R E N C E CP(2) capacitor CP (3) •> 400 pF position of CP(3): 342 cm capacitor CP(2) = 100 pF capacitor CP(6) = 87 pF UBE = 2000 Q CP (3) » CP (6) 16 LOOP 18 20 22 24 26 28 number of station 29 30 31 Fig. 69 Voltage phase along the transmission line; basic set-32 33 TUBE up Cf-90U f O O oo F i g . 71 RF contacts - 209 -Appendix A: CENTRE REGION CYCLOTRON RF SYSTEM PARAMETERS - THE RF FUNDAMENTAL 2 0 : 0 4 : 0 2 DEES 2 3 . 1 0 0 0 3 8 . 3 0 7. 00 - 210 -TRAP T R I T U M E 3 DATE 0 1 - 0 5 - 7 2 C ENTRE REGION 8 S E C T I O N S I N 2 ^ R E S O N A T O R S * RESONANT FREQUENCY C H A R A C T E R I S T I C IMPEDANCE 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 SHORT X=0 VOLTAGE TO CURRENT PHASE 90 HOT ARM LENGT H T I P TO T I P D I S T A N C E AVERAGE WIDTH OF S E C T I O N MAX.VOLTAGE ON RES#1 MAX. VOLTAGE ON RES#2 VOLTAGE PHASE S H I F T MAX.CURRENT IN RES#1 MAX.CURRENT IN RES#2 CURRENT D E N S I T Y AT ROOT POWER LOS S DUE TO BEAM POWER LOSS IN RESONATORS POWER LOSS IN RES#1 POWER LOSS I N R E S # 2 ^ C O U P L I N G LOOP* VOLTAGE INDUCED IN LOOP LOOP D I M E N S I O N S ARE HIGHT = LENGTH = WIDTH = T H I C K N E S S = LOOP P O S I T I O N LOOP S E L F INDUCTANCE CURRENT THROUGH LOOP POWER LOSS IN LOOP ' * T R A N S M I S S I O N L I N E * CHAR. IMPEDANCE OF L I N E = 4 9 . 7 8 OUTER D I A M E T E R = 1 1 . 7 5 0 INNER D I A M E T E R = 5 . 1 2 5 MAX.VOLTAGE WITHOUT C P ( 3 ) = 1 5 5 3 1 . MAX.CURRENT WITHOUT C P ( 3 ) = 3 1 2 . MAX.VOLTAGE WITH C P ( 3 ) = 2 2 8 0 2 . MAX.CURRENT WITH CP(3) = 4 5 8 . CU RRENT THROUGH C P ( 2 ) = 1 8 6 . 6 2 C A P A C I T O R A F T E R LOOP C P ( 2 ) = 1 5 0 . 0 0 0 P O S I T I O N OF C P ( 2 ) = 0 . 8 6 7 CURRENT THROUGH CP(3) = 6 0 0 . C A P A C I T O R C P ( 3 ) = 3 5 5 . 0 0 0 P O S I T I O N OF CP(3) = 3 . 8 2 0 0 L E N G T H A F T E R C P ( 3 ) '= 2 . 5 5 0 0 TOTAL L E N G T H OF L I N E = 6 . 3 7 0 0 VSWR -WITHOUT C P { 2 ) = 7.25 VSWR WITHOUT CP(3) = 2 0 . 1 2 VSWR A F T E R CP(3) = 4 2 . 9 3 POWER LOSS IN L I N E = 1 5 6 3 . POWER LOSS INNER C. = 1 0 8 8 . POWER LOSS OUTER C. = 4 7 5 . *POWER T U B E * TUBE CURRENT TUBE VOLTAGE CURRENT TO VOLTAGE PHASE C A P A C I T O R C P ( 6 ) AF T E R L I N E = CURRENT THROUGH C P ( 6 ) TOTAL POWER LOSS 3 . 0 8 6 2 2 0. 1 5 2 4 1 . 0 0 0 0 9 9 9 9 8 . 4 1 0 0 0 0 4 . 0 1 7 9 . 8 4 2 6 1 8 .8 2 6 1 9 . 0 2 6 . 188 5 0 0 0 . 1 2 0 3 1 3 . 5 7 4 7 9 . 5 7 8 3 4 . 9 2 7 7 . 2 1.75 17 .50 3.00 0.50 3.5 0 0 . 2 2 4 1 6 9 2 5 . 9 4 3 5 1 . 9 16 .62 1 4 6 6 9 . 9 0 .00 1 6 4 . 5 8 1 3 5 0 . 4 1 2 1 8 7 9 . MHZ OHMS P FARAD DEGREES METERS METERS METERS V O L T S V O L T S DEGREES AMPS AMPS AMPS/CM WATTS WATTS WATTS WATTS V O L T S I N C H E S I N C H E S I N C H E S I N C H E S I N C H E S MICROHENRY AMPS WATTS OHMS IN C H E S I N C H E S V O L T S AMPS VOLTS AMPS AMPS P F A R A D METERS AMPS P F A R A D METERS METERS METERS WATTS WATTS WATTS AMPS VOLTS DEGREES PFARAO AMPS WATTS ^ R E S O N A T O R S * AVERAGE MAGNETIC ENERGY 1 = AVERAGE E L E C T R I C ENERGY 1 = A V E R A G E 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 Q U A L I T Y MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA F O R E S H O R T E N I N G C O N D U C T I V I T Y S K I N DEPTH A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l = LUMPED INDUCTANCE AT V O l LUMPED C A P A C I T A N C E AT V 0 2 = LUMPED INDUCTANCE AT V 0 2 LUMPED C A P A C I T A N C E AT 101 = LUMPED INDUCTANCE AT 1 0 1 LUMPED C A P A C I T A N C E AT 102 = LUMPED INDUCTANCE AT 102 E Q U I V A L E N T VOLTAGE PEAK 1 = E Q U I V A L E N T VOLTAGE PEAK 2 = ION O R B I T I N G FREQUENCY A C C E L E R A T I N G T I M E P E R I O D TIME CONSTANT I N RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN QMEAS = SHUNT R E S I S T A N C E 1 AT VPEAK= 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= DEE TO DEE C A P A C I T A N C E A V E R A G E ENERGY IN C P ( 1 ) T 0 T = REACTANCE OF C T I P TOTAL 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 = R E S I S T A N C E DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA I MAG.COMPONENT 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 LOS S IN ROOT 1 POWER LOSS IN ROOT 2 MUTUAL INDUCTANCE 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 INDUCTANCE AT TUBE = C O U P L I N G C O E F F . A T TUBE 1 . 4 2 1 0 9 J O U L E S 1 . 4 2 1 0 9 J O U L E S 2 . 8 4 2 1 8 J O U L E S 1 . 4 2 1 2 5 J O U L E S 1 . 4 2 1 2 4 J O U L E S 2 . 8 4 2 4 9 J O U L E S 7 1 5 5 . 1 4 7 1 5 5 .14 7 4 9 9 . 5 2 2 3 . 7 1 6 0 MHZ 1 2 . 9 8 7 0 2 METERS 3 . 2 4 6 7 5 M E T E R S 1 4 5 . 1 4 1 4 3 1 0 * * 6 R A D / S E C 4 . 4 5 0 1 D E G R E E S 5 . 8 0 0 .10**7 MHOS 0 . 0 0 1 3 7 7 3 8 CM 0 . 0 0 0 0 3 2 6 8 1/M 5 6 8 . 4 6 P F A R A D 0 . 0 8 3 5 0 6 5 MICROHENRY 5 6 8 . 4 6 P F A R A D 0 . 0 8 3 5 0 6 2 MICROHENRY 9 1 6 . 3 6 P F A R A D 0 . 0 5 1 8 0 2 5 MICROHENRY 9 1 6 . 3 6 P F A R A D 0 . 0 5 1 8 0 2 8 MICROHENRY 1 0 2 1 5 0 . V O L T S 1 0 2 1 4 2 . V O L T S 4 . 6 2 0 0 MHZ 1.52 M I C R O S E C 98 .60 M I C R O S E C 3 5 . 0 1 2 0 3 1 3 . WATTS 8 6 9 8 5 . 4 4 OHMS 8 6 4 6 1 . 3 8 OHMS 3 5 7 . 6 7 9 9 OHMS 2 8 . 0 0 P F A R A D 0 . 2 8 0 0 0 6 2 J O U L E S - 2 4 6 . 0 7 OHMS 3 4 8 . 1 2 9 PFARAD/M 0 . 0 3 1 9 1 6 7 MICROHENRY/M 0 . 0 0 2 5 0 3 5 OHMS/M 0 . 1 0 1 6 M E T E R S 0 . 1 0 1 6 METERS 0 . 3 9 3 7 0 M E T E R S 0 . 6 2 7 1 1 9 M ET ER S**2 0 . 4 8 3 8 0 4 8 1/M - 0 . 0 0 2 5 8 7 3 OHMS 0 . 0 8 9 1 1 2 1 0 . 7 7 9 5 1 0 . 0 0 1 6 9 3 9 OHMS 174 4.44 WATTS 1 7 4 4 . 6 3 WATTS 0 . 0 0 7 7 4 6 8 MICROHENRY 0 . 0 5 6 6 2 0 4 3 0 . 0 0 6 207 2 MICROHENRY 0 . 0 1 1 8 3 9 0 MICROHENRY 0 . 0 0 1 9 7 9 8 9 - 212 -• C O U P L I N G LOOP* LOOP R E S I S T A N C E LOOP R E A C T A N C E LOOP AREA INPUT CURRENT D E N S I T Y LOOP S U R F A C E AVERAGE ENERGY IN LOOP LOSS DUE TO INPUT CURRENT = L O S S DUE TO C A V I T Y CURRENT = • T R A N S M I S S I O N L I N E * I MAG.COMPONENT OF ZTL A L F A IN L I N E D I S T R I B U T E D C A P A C I T Y D I S T R I B U T E D INDUCTANCE R E S I S T A N C E PER U N I T LENGTH = OUTER R A D I U S INNER R A D I U S AVERAGE E L E C T R I C ENERGY AVERAGE MAGNETIC ENERGY AVERAGE TOTAL ENERGY TL Q U A L I T Y FACTOR LUMPED C A P A C I T A N C E AT VTUBE= LUMPED INDUCTANCE AT VTUBE = S E R I E S R E S I S T A N C E OF L I N E = F I R S T S E C T I O N AVERAGE MAGNETIC ENERGY A V E R A G E E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 2 ) AVERAGE ENERGY IN C P ( 2 ) SECOND S E C T I O N AVERAGE MAGNETIC ENERGY A V E R A G E E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOS S TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 3 ) 1 N I T . V O L T A G E M A X . P O S I T I O N = T H I R D S E C T I O N AVERAGE MAGNETIC ENERGY A V E R A G E E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOS S OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T P O S I T I O N OF VOLTAGE MAXIMUM= AVERAGE ENERGY IN C P ( 3 ) *POWER T U B E * R E S I S T A N C E TO MATCH R E A C T A N C E TO MATCH REACTANCE OF C P ( 3 ) R E S I S T A N C E OF TUBE AVERAGE ENERGY IN C P ( 6 ) 0 . 0 0 8 7 6 2 3 3 2 . 5 3 6 2 2 0 . 0 1 9 7 5 8 3 . 4 0 4 0 . 0 8 1 2 9 0 0 . 0 0 0 0 3 7 7 2.95 3 4 8 . 9 3 - 0 . 0 0 4 5 4 3 1 0 . 0 0 0 0 4 4 1 5 6 6 . 9 5 6 9 0 . 1 6 5 9 4 4 0 . 0 0 4 3 9 6 0 . 1 4 9 2 2 4 8 0 . 0 6 5 0 8 7 4 0 . 0 2 9 5 5 4 5 0 . 0 2 9 5 5 5 0 0 . 0 5 9 1 0 9 5 5 4 8 9 . 8 3 0 . 1 1 0 8 6 4 2 8 . 1 7 7 7 3 1 1 . 3 2 0 3 0 . 0 0 0 0 8 7 7 0 . 0 0 1 1 9 5 8 0 . 0 0 1 2 8 3 5 1.4 3.1 4.5 0 . 7 5 7 - 4 5 . 9 3 0 . 0 0 2 7 5 5 4 0 . 0 0 9 9 5 5 6 0 . 0 0 1 9 9 7 3 0 . 0 1 1 9 5 2 9 1 6 0 . 1 3 6 6 . 9 5 2 7 . 0 0 . 9 0 5 - 1 9 . 4 1 5 . 3 1 7 0 . 0 1 9 4 7 4 0 0 . 0 0 2 7 3 1 6 0 . 0 2 2 2 0 5 6 3 1 3 . 2 7 1 8 . 1 1 0 3 1 . 2 0 . 9 5 4 8. 1 7 0 0 . 0 1 2 0 1 9 7 8 8 2 . 8 6 1 8 4 1 . 8 6 2 9 - 4 1 . 8 6 2 8 8 8 2 . 8 6 2 3 0 . 0 0 S 8 5 4 7 OHMS OHMS M E T E R S * * 2 AMPS/CM METER**2 J O U L E S WATTS WATTS OHMS 1/M P FARAD/M MICROHEMRY/M OHMS/M METERS METERS J O U L E S J O U L E S J O U L E S P F A R A D MICROHENRY OHMS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS J O U L E S J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS METERS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS METERS J O U L E S OHMS OHMS OHMS OHMS J O U L E S 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 0.6172 0.6172 0.6172 0.0 0.0 0 .0 0.0 0 .0 0.1734 0.1734 0.1734 0.1734 0.1734 0.0 0.1969 0.1969 0.1969 0 . 1969 0.1969 0.1969 0.1969 0.1969 0.1969 0.196 9 0.1969 0 . 196 9 0.1969 0.1969 0.1969 0.0 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.0 0.0 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) 0.0 -3 .93 -7 .16 -9 .08 -9 .28 -7 .54 -266.75 -252.88 -217.72 -164.23 -96 .87 -21 .28 -86.95 -32 .44 -25 .02 -868.91 -868.91 -974.06 -1072.37 -1163. 15 -1245.75 -1319.61 -1319.61 -1256.34 -1181 .69 -1096.34 -1001.06 -896.71 -784.24 -664.65 -539.04 -408.54 -274.33 -137.62 0.35 138.33 275.07 409.33 409.33 436.53 460.80 481.95 499.86 514.39 525.45 532.96. 536.86 537.13 533.77 526.78 516.23 502 .17 484.70 463.94 463.94 463.94 463.94 • ET ( K ) . •1 0.0 29509.2 56406.3 78310.5 93283.0 99998.4 100004.0 93288.3 78315.0 56409.8 29511.2 21.3 25599.3 4313.4 9277.2 9315.8 9315.8 9293.4 9207.0 9057.1 8844.9 8571.9 8571.9 7307.6 5980.5 4605.6 3203.7 1821.8 792.5 1512.8 2870.6 4274.2 5657.7 6997.7 8278.1 9485.9 10609.3 11637.6 11637.6 9989.3 8274.5 6505.6 4696.7 2867.1 1087.5 1065.6 2842.5 4672.1 6481.5 8251.0 9966 .6 11615.9 13187.2 14669.9 14669.9 14669.9 14669.9 I ( 11 K ) . . 0.0 0.02 0.06 0. 12 0. 20 0. 26 0. 26 2.30 4.15 5.65 6.68 7.14 6.77 7.11 -25 .94 -25 .94 -25 .94 -24 .38 -22 .66 -20 .77 -18 .74 -16 .58 12. 15 14.62 16.95 19.13 21.14 22.96 24.56 25.95 27. 10 28.01 28 .66 29.06 29. 19 29.06 28.66 28 .01 6.92 6.22 5.47 4.70 3.88 3.05 2. 19 1.31 0.43 -0 .46 -1 .35 - 2 . 2 2 -3 .08 -3 .93 - 4 . 7 4 - 5 . 53 0.0 16.62 16.62 . I ( 2 , K ) . . . -2618.82 -2502.92 -2165.47 -1636.33 -962.36 -203.20 -203.20 -962.40 -1636.41 -2165.57 -2503.04 -2618.96 -2532.23 -2616.54 -0 .07 -0 .07 -0 .07 -15 .68 -31 .18 -46 .46 -61 .42 -75 .94 -260.34 -275.34 -287.84 -297.74 -304.93 -309.36 -310.99 -309.80 -305.80 -299.03 -289.54 -277.44 -262.81 -245.81 -226.58 -205 .29 393.97 411.83 426.91 439.10 448.32 454.51 457.63 457.65 454.58 448.44 439.27 427. 12 412.09 394.27 373.79 350.78 -350.43 0.0 0.00 K K ) . . ; 2619. 2 50 3. 2165. 1636. 962. 203. 203. 962. 1636. 2166. 2503. 2619. 2532. 2617. 26. 26. 26. 29. 39. 51 . 64. 78. 261. 276. 288. 298. 306. 310. 312. 311. 307. 300. 291 . 279. 264. 248. 228. 207. 394. 412. 427. 439. 448. 455. 458. 458. 455 . 448. 439. 427. 412. 394. 374. 351. 350. 17. 17. - 214 -. P ( l f K ) . . . P ( 2 , K ) . . P ( K ) . . . 0.0 0.0 0. 0. 0.0 0.0 1 10288.39 0.0 10 2 8 8 . 1 0 2 8 8 . 84638.1 11.8 2 8560.88 0.0 8 5 6 1 . 1 8 8 4 9 . 168795.1 26.0 3 5704.00 0.0 5 7 0 4 . 2 4 5 5 3 . 249764.2 47.9 4 2706.86 0.0 2 7 0 7 . 2 7 2 6 0 . 319210.2 96.9 5 607.17 0.0 6 0 7 . 2 7 8 6 7 . 358831.2 4 9 2 . 1 6 . 2500.00 0.0 2 5 0 0 . 30367 . 358872. 3 -492.1 7 607.21 0.0 6 0 7 . 3 0 9 7 5 . 305630.6 -96.9 8 2707.12 0.0 2 7 0 7 . 3 3 6 8 2 . 196689.8 -47.9 9 5704.57 0.0 5 7 0 5 . 3 9 3 8 6 . 86266.8 -26.0 10 8561.80 0.0 8 5 6 2 . 4 7 9 4 8 . 19162.8 -11.8 11 10289.50 0.0 10 2 9 0 . 5 8 2 3 8 . 0.0 -0.4 12 0.0 0.0 0. 5 8 2 3 8 . 14009.0 - 1 0 . 1 13 0.0 0.0 0. 1 2 0 3 1 3 . 343.2 -1.6 14 2.95 0.0 3. 1 2 0 3 1 6 . 3 5 7 . 7 * * >V -J- .-L. -J* 15 0.0 0.0 0. 120316. 360. 6 3964.8 16 0.0 0.0 0. 1 2 0 3 1 6 . 360.6 3964.8 17 0.27 0.0 0. 120316. 358.9 -712.7 18 0.43 0.0 0. 12 0 3 1 6 . 352.3 -325.0 19 0.76 0.0 1. 120317. 340.9 -208.6 20 1.27 0.0 1. 1 2 0 3 1 8 . 325 . 1 -152.1 21 1 .93 0.0 2. 120320. 305.3 -118.3 22 0.0 0.0 0. 120 3 2 0 . 305.3 -33.1 23 31.17 0.0 3 1 . 1 2 0 3 5 1 . 221.9 -26.7 24 34.48 0.0 34 . 1 2 0 3 8 6 . 148.5 -20.9 25 37.29 0.0 3 7 . 120423. 88.1 -15.7 26 39.53 0.0 4 0 . 1 2 0 4 6 3 . 42.6 -10.8 27 41 .09 0.0 4 1 . 120504. 13.8 -6.5 2 8 41.94 0.0 4 2 . 120546 . 2.6 -11.5 29 42 .03 0.0 4 2 . 120588. 9.5 5.7 30 41.36 0.0 41 . 12 0 6 2 9 . 34.2 9.7 31 39.96 0.0 4 0 . 120669. 75.7 14.5 32 37.88 0.0 38 . 1 2 0 7 0 7 . 132.6 19.7 33 35.19 0.0 3 5 . 1 2 0 7 4 2 . 202.8 25.3 34 32.00 0.0 3 2 . 120774. 283.7 31.5 35 28.41 0.0 2 3 . 120802. 372.4 38.5 36 24.55 0.0 2 5 . 120827 . 465 .8 46.7 37 20 .57 0.0 2 1 . 120847. 560.3 56.5 38 0.0 0.0 0. 120847 . 560.3 -29.6 39 60 .75 0.0 6 1 . 120908. 412.7 -24.3 40 65.81 0.0 6 6 . 1 2 0 9 7 4 . 283.0 -19.4 41 70.15 0.0 7 0 . 121044. 174.8 -14.9 4 2 73.66 0.0 74. 1 2 1 1 1 8 . 91.1 -10.5 43 76 .23 0.0 7 6 . 121194. 33.9 -6.4 44 77.81 0.0 7 8 . 1 2 1 2 7 1 . 4.9 -2.7 45 78.35 0.0 7 8 . 121350. 4.7 2.7 46 77.82 0.0 7 8 . 1 2 1 4 2 8 . 33.3 6.4 47 76 .26 .0.0 7 6 . 121504. 8 9.8 10.5 48 73.70 0.0 7 4 . 1 2 1 5 7 8 . 172.8 14.8 49 70.20 0.0 7 0 . 121648. 279.8 19.4 50 65.87 0.0 6 6 . 121714. 4 0 8 . 1 24.2 51 60 .82 0.0 6 1 . 121774. 554.0 29.5 52 55.19 0.0 5 5 . 1 2 1 8 3 0 . 713.7 35.3 53 49.12 0.0 4 9 . 121879. 882.9 41.9 5 4 0.0 0.0 0. 121879. 0.0 41.9 55 0.0 0.0 0. 121879. 882. 9 0.0 56 0.0 0.0 0. 121879. 882.9 -0.0 57 STN ZSR. . . . . Z S I . . . . 1 0,0 0.0 2 0.0016 11 .7899 3 0.0040 26.0481 4 0.0092 47.8573 5 0.0294 96.9316 6 0.6749 492.1165 7 0.6749 -492.1443 8 0.0307 -96.9329 9 0.0116 -47.8577 10 0.0079 -26.0483 11 0.0073 -11.7901 12 0.0081 -0.0002 13 0.0073 -10.1093 14 0.0079 -1.6485 15 357.6799 0.0000 16 357.6887 32.5362 17 357.6887 32.5362 18 286.3120 -144.1833 19 161.9628 -175.5665 20 92.8928 -151.7822 21 58.3561 -124.7655 22 39.8273 -102.8337 23 3.5428 -32.6989 24 3.1661 -26 .3133 25 2.8960 -20.5378 26 2.7057 -15.1979 27 2.5787 -10.1590 28 2.5044 -5.3118 29 2.4773 -0.5620 30 2.4954 4.1776 31 2.5598 8.9934 32 2 .6755 13.9775 33 2.8517 19.2348 34 3.1033 24.8927 35 3.4545 31.1143 36 3.9435 38.1203 37 4.6331 46.2226 38 5.6303 55.8856 39 1.5567 -29.4937 40 1.4254 -24.2112 41 1.3273 -19.3354 42 1.2555 -14.7618 43 1.2051 -10.4062 44 1 .1733 -6.1980 45 1.1581 -2.0751 46 1.1588 2.0195 47 1.1752 6.1415 48 1.2084 10.3482 49 1.2602 14.7014 50 1.3336 19.2714 51 1.4334 24.1424 52 1.5666 29.4185 53 1.7437 35.2341 54 1.9806 41.7691 55 1.9806 41.7691 56 1.9806 41.7691 57 882.8623 -0 .0130 . .ZS PHASE VPHASE 0.0 90.0000 0.0 11.79 89.9919 359.9922 26.05 89.9911 359.9927 47.86 89.9890 359.9932 96.93 89.9825 359.9941 492.12 89.9214 359.9956 492.14 -89.9214 180.1528 96.93 -89.9818 180.1553 47.86 -89.9860 180.1593 26.05 -89.9827 180.1668 11.79 -89.9647 180.1881 0.01 -1 .1866 268.9695 10.11 -89.9586 180.1946 1.65 . -89.7247 180.4309 357.68 0.0000 180.1545 359.17 5.1975 185.3519 359.17 5.1975 185.3519 320.57 -26.7293 186.0163 238.86 -47.3080 186.6886 177.95 -58.5327 187.3785 137.74 -64.9332 188.0967 110.28 -68.8287 188.8556 32.89 -83 .8163 188.8556 26.50 -83.1389 189.8996 20.74 -81.9738 191.3961 15.44 -79.9051 193.7712 10.48 -75.7574 198.2080 5.87 -64.7571 209.4865 2.54 -12.7821 261.7341 4.87 59.1492 333.9375 9.35 74.1119 349.1768 14.23 79.1636 354.5151 19.45 81.5670 357.2207 25.09 82.8937 358.8730 31.31 83.6646 0.0024 38.32 84.0938 0.8355 46.45 84.2761 1.4857 56.17 84.2470 2.0157 29.53 -86.9786 2.0157 24.25 -86.6305 2.5046 19.38 -86.0728 .3 .1924 14.82 -85.1388 4.2485 10.48 -83.3941 6.1095 6.31 -79.2806 10.3354 2.38 -60.8345 28.8917 2.33 60.1530 149.9888 6.25 79.1670 169.1131 10.42 83.3394 173.3983 14.76 85.1007 175.2762 19.32 86.0414 176.3395 24.18 86.6021 177.0310 29.46 86.9517 177.5222 35.28 87.1668 177.8936 41.82 87.2.852 178. 1877 41.82 87.2852 0.0 41.82 87.2852 0.0 882.86 -0 .0008 0.0 - 216 -T R A P T R I T U N E 3 DATE 0 1 - 0 5 - 7 2 2 0 : 0 9 : 5 0 CENTRE R E G I O N 8 S E C T I O N S IN 2 D E E S • R E S O N A T O R S * RESONANT FREQUENCY = 2 3 . 1 0 0 0 C H A R A C T E R I S T I C IMPEDANCE = 3 8 . 3 0 T I P TO T I P C A P A C I T A N C E = 7 . 0 0 V O L T A G E , C U R R E N T PEAKS,POWER L O S S RMS AT SHORT X=0 VOLTAGE TO CURRENT PHASE 9 0 HOT ARM L E N G T H = 3 . 0 8 6 2 2 T I P TO T I P D I S T A N C E = 0 . 1 5 2 4 AVERAGE WIDTH OF S E C T I O N = 1 . 0 0 0 0 MAX.VOLTAGE ON RES#1 = 9 9 9 9 8 . 4 MAX. VOLTAGE ON RES#2 = 1 0 0 0 0 4 . 0 VOLTAGE PHASE S H I F T = 1 7 9 . 8 4 MAX.CURRENT IN RES#1 = 2 6 1 8 . 8 MAX.CURRENT IN RES#2 = 2 6 1 9 . 0 CURRENT D E N S I T Y AT ROOT = 2 6 . 1 8 8 POWER LOSS DUE TO BEAM = 0. POWER LOSS IN RESONATORS = 1 1 5 3 1 3 . POWER LOSS IN RES#1 = 5 7 4 7 9 . POWER LOSS INRES# 2 = 5 7 8 3 4 . • C O U P L I N G LOOP* VOLTAGE INDUCED IN LOOP = 9 2 7 7 . 2 LOOP D I M E N S I O N S ARE HIGHT = 1.75 L E N G T H = 1 7 . 5 0 WIDTH = 3.00 T H I C K N E S S = 0.50 LOOP P O S I T I O N = 3 . 5 0 LOOP S E L F INDUCTANCE = 0 . 2 2 4 1 6 9 CURRENT THROUGH LOOP = 2 4 . 8 6 POWER LOSS IN LOOP = 3 5 1 . 6 • T R A N S M I S S I O N L I N E * CHAR. IMPEDANCE OF L I N E = 4 9 . 7 8 OUTER D I A M E T E R = 1 1 . 7 5 0 INNER D I A M E T E R = 5 . 1 2 5 MAX.VOLTAGE WITHOUT C P ( 3 ) = 1 5 5 2 8 . MAX.CURRENT WITHOUT C P ( 3 ) = 3 1 2 . MAX.VOLTAGE WITH C P ( 3 ) = 2 2 8 0 2 . MAX.CURRENT WITH C P ( 3 ) = 4 5 8 . CURRENT THROUGH C P ( 2 ) = 1 8 6 . 4 5 C A P A C I T O R A F T E R LOOP C P ( 2 ) = 1 5 0 . 0 0 0 P O S I T ION OF C P ( 2 ) • = 0 . 8 6 7 CURRENT THROUGH C P ( 3 ) = 6 0 0 . C A P A C I T O R C P ( 3 ) = 3 5 5 . 0 0 0 P O S I T I O N OF C P ( 3 ) = 3 . 8 2 0 0 LENGTH A F T E R C P ( 3 ) = 2 . 5 5 0 0 TOTAL L E N G T H OF L I N E = 6 . 3 7 0 0 VSWR WITHOUT C P ( 2 ) = 7.55 VSWR WITHOUT C P ( 3 ) = 2 0 . 9 8 VSWR A F T E R C P ( 3 ) = 4 4 . 7 7 POWER LOSS IN L I N E = 1 5 6 2 . POWER LOSS INNER C. = 1 0 8 8 . POWER LOSS OUTER C. = 4 7 4 . •POWER T U B E * TUBE CURRENT = 1 5 . 9 4 TUBE VOLTAGE = 1 4 6 6 9 . 1 CURRENT TO VOLTAGE PHASE = - 0 . 1 7 C A P A C I T O R C P ( 6 ) A F T E R L I N E = 1 6 4 . 5 8 1 CURRENT THROUGH C P ( 6 ) = 3 5 0 . 4 TOTAL POWER L O S S = 1 1 6 8 7 8 . MHZ OHMS P F A R A D D E G R E E S METERS METERS METERS V O L T S VOLT'S D E G R E E S AMPS AMPS AMPS/CM WATTS WATTS WATTS WATTS V O L T S I N C H E S I N C H E S I N C H E S I N C H E S I N C H E S MICROHENRY AMPS WATTS OHMS I N C H E S I N C H E S V O L T S AMPS V O L T S AMPS AMPS P F A R A D METERS AMPS P F A R A D METERS M E T E R S METERS WATTS WATTS WATTS AMPS V O L T S D E G R E E S P F A R A D AMPS WATTS - 217 -• R E S O N A T O R S * AVERAGE MAGNETIC ENERGY 1 = A V E R A G E 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 COMPUTED Q U A L I T Y MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA F O R E S H O R T E N I N G C O N D U C T I V I T Y S K I N DEPTH A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l = LUMPED INDUCTANCE AT V O l LUMPED C A P A C I T A N C E AT V02 = LUMPED INDUCTANCE AT V 0 2 LUMPED C A P A C I T A N C E AT 101 = LUMPED INDUCTANCE AT 101 LUMPED C A P A C I T A N C E AT 102 = LUMPED INDUCTANCE AT 102 E Q U I V A L E N T VOLTAGE PEAK 1 = E Q U I V A L E N T VOLTAGE PEAK 2 = I ON O R B I T I N G FREQUENCY A C C E L E R A T I N G T I M E P E R I O D T I M E CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN QMEAS = 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 VPEAK= SHUNT R E S I S T A N C E AT VLOOP= DEE TO DEE 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 TOTAL 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 = R E S I S T A N C E DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA B E T A IMAG.COMPONENT OF ZRES 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 LOSS IN ROOT 1 POWER LOSS IN ROOT 2 MUTUAL INDUCTANCE 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 INDUCTANCE AT TUBE = C O U P L I N G C O E F F . A T TUBE 1 . 4 2 1 0 9 1 . 4 2 1 0 9 2.84 218 1 . 4 2 1 2 5 1 . 4 2 1 2 4 2 . 8 4 2 4 9 7 1 5 5 .14 7 1 5 5 . 1 4 7 4 9 9 . 5 2 2 3 . 7 1 6 0 1 2 . 9 8 7 0 2 3 . 2 4 6 7 5 1 4 5 . 1 4 1 4 3 4 . 4 5 0 1 5 . 8 0 0 0 . 0 0 1 3 7 7 3 8 0 . 0 0 0 0 3 2 6 8 5 6 8 . 4 6 0 . 0 8 3 5 0 6 5 5 6 8 . 4 6 0 . 0 8 3 5 0 6 2 9 1 6 . 3 6 0 . 0 5 1 8 0 2 5 9 1 6 . 3 6 0 . 0 5 1 8 0 2 8 9 9 9 9 8 . 1 0 0 0 0 4 . 4 . 6 2 0 0 0.0 9 8 . 6 0 0.0 1 1 5 3 1 3 . 8 6 9 8 5 . 4 4 8 6 4 6 1 . 3 8 3 7 3 . 1 8 9 0 2 8 . 00 0 . 2 8 0 0 0 6 2 -24 6.07 3 4 8 . 1 2 9 0 . 0 3 1 9 1 6 7 0 . 0 0 2 5 0 3 5 0 . 1 0 1 6 0 . 1 0 1 6 0 . 3 9 3 7 0 0 . 6 2 7 1 1 9 0 . 4 8 3 8 0 4 8 - 0 . 0 0 2 5 8 7 3 0 . 0 8 9 1 1 2 1 0 . 7 7 9 5 1 0 . 0 0 1 6 9 3 9 1 7 4 4 . 4 4 1 7 4 4 . 6 3 0 . 0 0 7 7 4 6 8 0 . 0 5 6 6 2 0 4 3 0 . 0 0 6 2 0 7 2 0 . 0 1 2 0 8 5 6 0 . 0 0 1 9 3 8 6 7 J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S MHZ -METERS METERS 1 0 * * 6 R A D / S E C D E G R E E S 1 0 * * 7 MHOS CM 1/M P F A R A D MICROHENRY P F A R A D MICROHENRY P F A R A D MICROHENRY P F A R A D MICROHENRY V O L T S V O L T S MHZ M I C R O S E C M I C R O S E C WATTS OHMS OHMS OHMS P F A R A D J O U L E S OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS METER S**2 1/M OHMS OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY - 218 -• C O U P L I N G L O O P * LOOP R E S I S T A N C E LOOP R E A C T A N C E LOOP AREA INPUT CURRENT D E N S I T Y LOOP S U R F A C E AVERAGE ENERGY IN LOOP L O S S DUE TO INPUT CURRENT = LOSS DUE TO C A V I T Y CURRENT = •-TRANSMISSION L I N E * IMAG .COMPONENT OF ZTL A L F A IN L I N E D I S T R I B U T E D C A P A C I T Y D I S T R I B U T E D INDUCTANCE R E S I S T A N C E PER UNIT L E N G T H = OUTER R A D I U S INNER R A D I U S A V E R A G E E L E C T R I C ENERGY AVERAGE MAGNETIC ENERGY AVERAGE TOTAL ENERGY TL Q U A L I T Y FACTOR LUMPED C A P A C I T A N C E AT VTUBE= LUMPED INDUCTANCE AT VTUBE = S E R I E S R E S I S T A N C E OF L I N E = F I R S T S E C T I O N A V E R A G E MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOS S INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T REACTANCE OF C P ( 2 ) AVERAGE ENERGY IN C P ( 2 ) SECOND S E C T I O N A V E R A G E MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY A V E R A G E TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOS S INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T REACTANCE OF C P ( 3 ) IN I T . V O L T A G E M A X . P O S I T I O N = TH I R D S E C T I O N A V E R A G E MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY A V E R A G E TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T P O S I T I O N OF VOLTAGE MAXIMUM= AV E R A G E ENERGY IN C P ( 3 ) *POWER T U B E * R E S I S T A N C E TO MATCH REACTANCE 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 TUBE A V E R A G E ENERGY IN C P ( 6 ) 0 . 0 0 8 7 6 2 3 3 2 . 5 3 6 2 2 0 . 0 1 9 7 5 8 3 . 2 6 2 0 . 0 8 1 2 9 0 0 . 0 0 0 0 3 4 6 2.71 3 4 8 . 9 3 - 0 . 0 0 4 5 4 3 1 0 . 0 0 0 0 4 4 1 5 66 . 9 5 6 9 0 . 1 6 5 9 4 4 0 . 0 0 4 3 9 6 0 . 1 4 9 2 2 4 8 0 . 0 6 5 0 8 7 4 0 . 0 2 9 5 4 2 7 0 . 0 2 9 5 4 4 4 0 . 0 5 9 0 8 7 1 5 4 8 9 . 2 9 0. 1 0 2 0 0 4 6 5 . 3 8 0 8 6 1 2 . 3 0 5 1 0 . 0 0 0 0 8 6 3 0 . 0 0 1 1 9 4 4 0 . 0 0 1 2 8 0 7 1.3 3.1 4.4 0 . 7 6 6 - 4 5 .93 0 . 0 0 2 7 5 0 3 0 . 0 0 9 9 4 9 8 0 . 0 0 1 9 9 5 0 0 . 0 1 1 9 4 4 8 1 6 0 . 0 3 6 6 .7 52 6 .7 0 . 9 0 9 - 1 9 . 4 1 5 .317 0 . 0 1 9 4 7 3 6 0 . 0 0 2 7 3 0 7 0 . 0 2 2 2 0 4 3 3 1 3 . 2 7 1 8 . 0 1 0 3 1 . 2 0 . 9 5 6 8. 1 7 0 0 . 0 1 2 0 1 8 6 9 2 0 . 5 3 9 1 4 1 . 3 5 7 3 - 4 1 . 8 6 2 9 9 2 0 . 5 3 2 0 0 . 0 0 8 8 5 3 7 OHMS OHMS METERS**2. AMPS/CM METER**2 J O U L E S WATTS WATTS OHMS 1/M PFARAD/M MICROHENRY/M OHMS/M METERS METERS J O U L E S J O U L E S J O U L E S P F A R A D MICROHENRY OHMS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS J O U L E S J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS METERS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS METERS J O U L E S OHMS OHMS OHMS OHMS J O U L E S STN. I 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 3 3 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 0.6172 0.6172 0.6172 0.0 0.0 0.0 0.0 0.0 0.1734 0.1734 0.1734 0.1734 0.1734 0.0 0 . 1969 0 .1969 0.196 9 0.1969 0.196 9 0.1969 0 . 196 9 0.1969 0. 196 9 0.1969 0.1969 0.1969 0. 196 9 0 .1969 0.1969 0.0 0.1700 0.1700 0.1700 0.1700 0.1700 0 .1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.1700 0.0 0 .0 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 E (2,K 0. -3 . - 7 . - 9 . - 9 . - 7 . - 2 6 6 . -252 . - 2 1 7 . -164 . - 9 6 . -21 . - 8 6 . - 3 2 . - 2 5 . - 8 3 3 . - 8 3 3 . -934 . -1028. -1115. -1194. -1265. -1265. -1204. -1133. -1051 . - 9 5 9 . - 859 . -751 . -636 . - 516 . -391 . - 2 6 2 . -131 . 1 . 133 . 264. 393. 393 . 419. 442. 462. 479 . 493. 503 . 510. 514. 514. 511 . 504. 494. 480. 463 . 443 . 443 . 443. 443 . 219 -) . . .ET(K) . . 0.0 29509.2 56406.3 78310.5 93283.0 99998 .4 75 100004.0 88 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 0 93 16 08 28 54 72 23 87 28 95 44 02 84 84 62 83 82 97 72 72 95 26 31 84 68 72 95 40 17 38 21 17 54 73 53 53 45 56 68 68 45 88 91 48 5 7 17 32 04 41 52 47 47 4 7 47 I ( 1,K) . . 0.0 0.02 0.06 0.12 0. 20 0. 26 0. 26 2. 30 4.15 5.65 6.68 7.14 6.77 7.11 -24.86 -24 .86 -24 .86 -23 .37 -21 .71 -19 .90 -17.96 -15.88 11.67 14.04 16. 28 18.37 20. 29 22.04 23. 58 24.91 26.01 26.88 27. 51 27.88 28.01 27.88 27. 50 26.87 6. 59 5.92 5.21 4.46 3.63 2.88 2.05 1.21 0. 36 -0 .49 - 1 . 33 - 2 . 17 - 3 . 0 0 - 3 . 8 0 - 4 . 58 - 5 . 3 3 0.0 15.94 15.94 I ( 2 , K ) . . . -2618.82 -2502.92 -2165.47 -1636.33 -962.36 -203.20 -203.20 -962.40 -1636.41 -2165.57 -2503.04 -2618.96 -2532.23 -2616.54 -0 .07 -0 .07 -0 .07 -15 .68 -31.18 -46 .46 -61 .42 -75 .94 -260.34 -275.34 -287.84 -297.74 -304.93 -309.37 -310.99 -309.80 -305.30 -299.03 -289.55 -277.44 -262.82 -245.81 -226.58 -205.29 393.97 411.83 426.90 439.09 448.32 454.51 457.63 457.65 454.58 448.44 439.26 427.12 412.09 394.27 373.79 350.78 -350.45 0.0 -0 .05 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. 0.0 0.0 1 10288.39 0.0 10288. 10288. 84638.1 11.8 2 8560.88 0 . 0 8561. 18849. 168795.1 26.0 3 5704.00 C O 5704. 24553. 249764.2 47.9 4 • 2706.86 o .d 27 07. 27260. 319210.2 96.9 5 607.17 0 .0 607. 27867. 358831.2 492.1 6 0.0 0.0 0 . 27867. 358872.3 -492.1 7 607.21 0 .0 607. 28475. 305630.6 -96 .9 8 2707.12 0 .0 2707. 31182. 196689.8 -47 .9 9 5704.57 0.0 5705. 36886. 86266.8 -26 .0 10 8561.80 0 .0 8562. 45448. 19162.8 -11 .8 11 10289.50 0.0 10290. 55738. 0.0 - 0 . 4 12 0 .0 0.0 0. 55738. 14009.0 -10 .1 13 0 .0 0.0 0. 115313. 343.2 - 1 . 6 14 2.71 0.0 3 . 115316. 373.2* 15 0 .0 0 .0 0 . 115316. 376.0 4313.2 16 0.0 0 .0 0. 115316. 376.0 4313.2 17 0 .25 0 .0 0 . 115316. 374.2 -701.2 18 0.41 0 .0 0. 115316. 367.2 -322 .2 19 0 .75 0.0 1. 115317. 355.2 -207.3 20 1 .25 0.0 1. 115318. 338.7 -151.3 21 1.92 0 .0 2 . 115320. 318.0 -117.7 22 0 .0 0 .0 0. 115320. 318.0 -33 .0 23 . 31.17 0 .0 31 . 115351. 230.9 -26 .7 24 34.47 0.0 34 . 115386. 154.5 -20 .9 25 37.28 0.0 37 . 115423. 91.5 -15 .6 26 39.51 0.0 40 . 115462. 44. 1 -10 .7 27 41.08 0.0 4 1 . 115503. 14. 1 - 6 . 4 28 41 .92 0.0 42 . 115545. 2.5 -10 .9 29 42.01 0.0 42 . 115587. 9.7 5.6 30 41 .34 0.0 4 1 . 115629. 35.5 9.7 31 39.94 0.0 40 . 115668. 78.9 14. 5 32 37.86 0 .0 38. 115706. 138. 3 19.6 33 35. 17 0.0 35 . 115741. 211.5 25.3 34 31 .97 0 .0 32. 115773. 296.0 31.5 35 28.38 0.0 28. 115802. 388 .5 38.5 36 24.52 0.0 25. 115826. 485.9 46.7 37 20.54 0.0 2 1 . 115847. 584.5 56.5 38 0.0 0 .0 0. 115847. 584. 5 -29 .6 39 60.75 0.0 61 . 115907. 430.4 -24 .3 40 65 .81 0.0 66 . 115973. 295.1 -19 .4 41 70.15 0.0 70 . 116043. 182.3 -14 .9 42 73 .65 0.0 74. 116117. 94.9 -10 .5 43 76.23 0.0 76 . 116193. 35.3 - 6 . 4 44 77.81 0.0 78 . 116271. 5.0 - 2 . 7 45 78.34 0.0 78. 116349. 4.8 2.6 46 77.82 0.0 78. 116427. 34.6 6.3 4 7 76.26 0.0 76. 116503. 93.6 10.5 48 73 .70 0.0 74. 116577. 180.1 14.8 49 70.20 0.0 70 . 116647. 291 .7 19.4 50 65 .87 0.0 66. 116713. 425.4 24.2 51 60.82 0.0 6 1 . 116774. 577 .6 29.5 52 55.19 0.0 55. 116829. 744.2 35.3 53 49.12 0.0 49 . 13.6878. 920.5 . 41.9 54 0.0 0 .0 0. 116878. 0.0 41.9 55 0.0 0.0 0 . 116878. 920.5 0.0 56 0.0 0.0 0. 116878. 920.5 2.7 57 - 221 -Z S . . . . . . . V P H A SE 1 0.0 0.0 0.0 90.0000 0.0 2 0.0016 11 .7899 11.79 89.9919 359.9922 3 0.0040 26 .0481 26.05 89.9911 359.9927 4 0.0092 47.8573 47.86 89.9890 359.9932 5 0.0294 96 .9316 96 .93 89.9825 359.9941 6 0.6749 492.1165 492.12 89.9214 359.9956 7 0 .6749 -492.1443 492.14 -89.9214 180.1528 8 0.0307 -96.9329 96.93 -89.9818 180.1553 9 0.0116 -47.8577 47.86 -89.9860 180.1593 10 0.0079 -26.0483 26.05 -89.9827 180.1668 11 0 .0073 -11 .7 901 11.79 -89.9647 180.1881 12 0.0081 -0 .0002 0.01 - 1 . 1866 268.9695 13 0.0073 -10.1093 10. 11 -89.9586 180.1946 14 0.0079 -1 .6485 1.65 -89.7247 180.4309 15 373.1890 0.0000 373. 19 0.0000 180.1545 16 373.1975 32.5362 374.61 4.9826 185.1371 17 373.1975 32 .5362 3 74.61 4.9826 185.1371 18 291.2400 -155 .4069 330.11 -28.0846 185.7743 19 159.7585 -182 .0290 242.19 -48.7280 186.4193 20 90.2712 -154.6522 179.07 -59.7276 187.0813 21 56 .3285 -126.1115 138 .12 -65.9317 187.7706 22 38.3177 -103 .5196 110.38 -69.6880 188.4993 23 3.3961 -32.6862 32.86 -84.0682 188.4993 24 3.0352 -26.3003 26.47 -83.4168 189.5020 25 2 .7764 -20.5246 20.71 -82.2962 190.9404 26 2.5942 -15 .1844 15.40 -80.3048 193.2255 27 2.4725 -10.1451 10.44 -76.3031 197.5044 28 2.4015 -5 .2974 5.82 -65.6134 208.4609 29 2 .3757 -0.5468 2.44 -12.9608 261.3748 30 2.3932 4.1939 4.83 60.2896 334.8862 31 2 .4552 9.0112 9.34 74.7591 349.6211 32 2.5664 13.9971 14.23 79.6101 354.7468 33 2 .7356 19.2569 19.45 81.9148 357.3416 34 2.9773 24.9181 25. 10 83.1864 358.9255 35 3 .3146 31.1443 31.32 83.9250 0.0081 36 3.7843 38.1567 38.34 84.3360 0.8066 37 4 .4468 46.2684 46.48 84.5101 1.4298 38 5 .4050 55 .9456 56.21 84.4816 1.9379 39 1.4924 -29.4963 29.53 -87.1035 1.9379 40 1.3665 -24.2133 24.25 -86.7697 2.4067 41 1.2725 -19.3371 19.38 -86.2350 . 3.0662 42 1.2036 -14.7632 14.81 -85.3391 4.0792 43 1 .1554 -10.4073 10.47 -83.6651 5.8646 44 1.1249 -6 .1989 6.30 -79.7146 9.9229 45 1 .1104 -2 .0759 2.35 -61.8578 27.8853 46 1.1110 2.0189 2.30 61.1758 151.0240 47 1 .1268 6.1412 6.24 79.6025 169.5565 48 1.1587 10.3480 10.41 83.6111 173.6732 49 1 .2083 14.7014 14.75 85.3012 175.4752 50 1.2788 19.2717 19.31 86.2036 176.4952 51 1 .3745 24.1430 24 . 18 86.7415 177.1584 52 1.5023 29.4194 29.46 87.0768 177.6295 53 1.6721 35.2355 35 .28 87.2830 177.9856 54 1.8994 41.7711 41.81 87.396 5 178.2676 55 1 .8994 41 .7711 41.81 87.3965 0.0 56 1.8994 41 .7711 41.81 87.3965 0.0 57 920.5320 2.6655 920.54 0.1659 0.0 - 222 -Appendix B: CENTRE REGION CYCLOTRON RF SYSTEM PARAMETERS - THE RF THIRD HARMONIC - 223 -TRAP T R I T U N E 3 DATE 0 2 - 2 6 - 7 2 1 3 : 0 9 : 3 5 CENTRE R E G I O N 8 S E C T I O N S IN 2 D E E S • R E S O N A T O R S * RESONANT FREQUENCY = 6 9 . 3 0 0 0 C H A R A C T E R I S T I C IMPEDANCE = 3 8 . 3 0 T I P TO T I P C A P A C I T A N C E = 7.11 VO L T A G E , C U R R E N T PEAKS,POWER LOSS RMS AT SHORT X=0 VOLTAGE TO CURRENT PHASE 9 0 HOT ARM LENGT H = 3 . 0 8 6 2 2 T I P TO T I P D I S T A N C E = 0 . 1 5 2 4 AVERAGE WIDTH OF S E C T I O N = 1 . 0 0 0 0 MAX.VOLTAGE ON RES#1 = 1 2 9 9 2 . 5 MAX. VOLTAGE ON RES#2 = 1 3 0 0 8 . 0 VOLTAGE PHASE S H I F T = 1 7 9 . 9 1 MAX.CURRENT IN RES#1 = 3 4 8 . 7 MAX.CURRENT IN RES#2 = 3 4 9 . 0 CURRENT D E N S I T Y AT ROOT = 3 . 4 8 7 POWER LOSS DUE TO BEAM = 0. POWER LOSS IN RESONATORS = 3 5 3 2 . POWER LOSS IN RES#1 = 1 7 6 2 . POWER LOS S I N R E S # 2 = 1 7 7 1 . • C O U P L I N G LOOP* VOLTAGE INDUCED IN LOOP = 1 2 4 9 . 1 LOOP D I M E N S I O N S ARE HIGHT = 1.75 LENGTH = 6, WIDTH = 3. T H I C K N E S S = LOOP P O S I T I O N LOOP S E L F INDUCTANCE = 0, CURRENT THROUGH LOOP POWER LOSS I N LOOP • T R A N S M I S S I O N L I N E * IMPEDANCE OF L I N E D I A M E T E R D IAMETER MAX.VOLTAGE WITHOUT C P ( 3 ) = MAX.CURRENT 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 ) C A P A C I T O R C P ( 3 ) P O S I T I O N OF C P ( 3 ) C P < 3 ) OF L I N E CP ( 2 ) C P ( 3 ) C P ( 3 ) I N L I N E CHAR. OUTER INNER LENGTH A F T E R TOTAL LENGTH VSWR WITHOUT VSWR WITHOUT VSWR A F T E R POWER LOSS POWER LOSS INNER C. POWER LOSS OUTER C. •-POWER T U B E * TUBE CURRENT TUBE VOLTAGE CURRENT TO VOLTAGE PHASE C A P A C I T O R C P ( 6 ) A F T E R L I N E = CURRENT THROUGH C P ( 6 ) . TOTAL POWER LOS S 00 00 0.50 3.50 0 8 6 6 7 1 5.66 5.0 4 9 . 9 7 5 . 9 8 0 2 . 6 0 0 2 4 2 6 . 4 9 . 2 4 2 6 . 4 9 . 3 4 . 0 6 65 . 0 0 0 0. 2 5 4 0. 0 . 0 0 0 4 . 1 0 5 0 1 . 0 0 0 0 5. 1 0 5 0 4 . 5 6 16 .66 1 6 . 4 6 4 7 . 3 3 . 1 4 . 1 1 . 9 7 5 9 8 . 3 - 0 . 0 0 1 7 5 . 0 4 4 45 .6 3 5 8 0 . M H Z OHMS P F A R A D D E G R E E S METERS -METERS METERS V O L T S V O L T S D E G R E E S AMPS AMPS AMPS/CM WATTS WATTS WATTS WATTS V O L T S I N C H E S I N C H E S I N C H E S I N C H E S I N C H E S MICROHENRY AMPS WATTS OHMS • I N C H E S I N C H E S V O L T S AMPS V O L T S AMPS AMPS P F A R A D METERS AMPS P F A R A D METERS METERS METERS WATTS WATTS WATTS AMPS V O L T S D E G R E E S P F A R A D AMPS WATTS • R E S O N A T O R S * - 224 -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 Q U A L I T Y MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA F O R E S H O R T E N I N G C O N D U C T I V I T Y S K I N DEPTH A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l = LUMPED INDUCTANCE AT V O l LUMPED C A P A C I T A N C E AT V 0 2 = LUMPED INDUCTANCE AT V 0 2 LUMPED C A P A C I T A N C E AT 101 = LUMPED INDUCTANCE AT 101 LUMPED C A P A C I T A N C E AT 1 0 2 = LUMPED INDUCTANCE AT 102 E Q U I V A L E N T VOLTAGE PEAK 1 = E Q U I V A L E N T VOLTAGE PEAK 2 = ION O R B I T I N G FREQUENCY A C C E L E R A T I N G T I M E P E R I O D T I M E CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN OMEAS = 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 VPEAK= SHUNT R E S I S T A N C E AT VLOOP= DEE TO DEE C A P A C I T A N C E AVERAGE ENERGY IN C P I D T O T = REACTANCE OF C T I P TOTAL 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 = R E S I S T A N C E DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA . = BETA IMAG.COMPONENT OF ZRES 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 LOSS I N ROOT 1 POWER LOSS IN ROOT 2 MUTUAL INDUCTANCE 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 INDUCTANCE AT TUBE = C O U P L I N G C O E F F . A T TUBE. 0 . 0 2 5 1 5 0 . 0 2 5 1 5 0 . 0 5 0 2 9 0 . 0 2 5 2 1 0 . 0 2 5 2 0 0 . 0 5 0 4 1 1 2 4 1 3 . 2 9 1 2 4 1 3 . 2 9 1 2 9 8 9 . 5 4 7 1 . 1 4 8 1 4 . 3 2 9 0 1 1 . 0 8 2 2 5 4 3 5 . 4 2 4 3 2 1 3 . 3 5 0 1 5 . 8 0 0 0 . 0 0 0 7 9 5 2 3 0 . 0 0 0 0 5 6 6 1 5 9 5 . 9 2 0 . 0 0 8 8 5 0 9 5 9 5 . 9 0 0 . 0 0 8 8 5 1 1 1 0 1 . 9 8 0 . 0 5 1 7 2 1 1 1 0 1 . 9 7 0 . 0 5 1 7 2 6 6 1 2 9 9 2 . 1 3 0 0 8 . 4 . 6 2 0 0 0.0 5 7 . 0 2 0.0 3 5 3 2 . 4 7 9 0 4 . 4 8 4 7 7 8 4 . 6 4 2 2 0 . 8 3 0 8 2 8 . 4 4 0 . 0 0 4 8 0 7 2 - 8 0 . 7 4 3 4 8 . 1 2 9 0 . 0 3 1 9 1 6 7 0 . 0 0 4 3 3 6 2 0 . 1 0 1 6 0. 1 0 1 6 0 . 3 9 3 7 0 0 . 6 2 7 1 2 0 1 . 4 5 1 4 1 4 1 - 0 . 0 0 1 4 9 3 8 0 . 0 3 0 5 5 3 1 0 . 4 1 4 3 2 0 . 0 0 0 3 1 0 5 5 3 . 5 5 5 3 . 6 7 0 . 0 0 0 8 4 9 9 0 . 0 3 0 6 8 5 0 0 0 . 0 0 2 1 2 8 2 0 . 0 0 0 4 0 2 0 0 . 0 0 1 5 7 7 9 8 J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S MHZ METERS METERS 1 0 * * 6 R A D / S E C D E G R E E S 1 0 * * 7 MHOS CM 1/M P F A R A D MICROHENRY P F A R A D MICROHENRY P F A R A D MICROHENRY P F A R A D MICROHENRY V O L T S V O L T S MHZ M I C R O S E C M I C R O S E C WATTS OHMS OHMS OHMS P F A R A D J O U L E S 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 -• C O U P L I N G L O O P * LOOP R E S I S T A N C E LOOP REACTANCE LOOP AREA INPUT CURRENT D E N S I T Y LOOP S U R F A C E AVERAGE ENERGY IN LOOP LOSS DUE TO INPUT CURRENT = LOSS DUE TO C A V I T Y CURRENT = • T R A N S M I S S I O N L I N E * IMAG.COMPONENT OF ZTL A L F A IN L I N E D I S T R I B U T E D C A P A C I T Y D I S T R I B U T E D INDUCTANCE R E S I S T A N C E PER U N I T LENGTH = OUTER R A D I U S INNER R A D I U S AVERAGE E L E C T R I C ENERGY AVERAGE MAGNETIC ENERGY AVERAGE TOTAL ENERGY TL Q U A L I T Y FACTOR LUMPED C A P A C I T A N C E AT VTUBE= LUMPED INDUCTANCE AT VTUBE = S E R I E S R E S I S T A N C E OF L I N E = F I R S T S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P { 2 ) AVERAGE ENERGY IN C P ( 2 ) SECOND S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 3 ) I N I T . V O L T A G E M A X . P O S I T I O N = T H I R D S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T P O S I T I O N OF VOLTAGE MAXIMUM= AVERAGE ENERGY I N C P ( 3 ) •POWER T U B E * R E S I S T A N C E TO MATCH REACTANCE TU MATCH REA C T A N C E OF C P ( 3 ) RES I S T A N C E OF TUBE . . = AVERAGE ENERGY I N C P ( 6 ) 0 . 0 0 6 8 6 5 7 3 7 . 7 3 8 6 6 0 . 0 0 6 7 7 4 0 . 7 4 2 0 . 0 3 6 7 7 4 0 . 0 0 0 0 0 0 7 0 . 1 1 4 . 8 5 - 0 . 0 0 5 1 6 5 2 0 . 0 0 0 1 5 0 0 1 6 6 . 7 0 0 7 0. 1 6 6 5 8 2 0 . 0 1 4 9 9 4 0 . 0 7 5 9 4 6 0 0 . 0 3 3 0 2 0 0 0 . 0 0 0 2 6 2 4 0 . 0 0 0 2 6 2 5 0 . 0 0 0 5 2 4 9 4 8 5 0 . 1 2 0 . 7 1 9 2 5 7 . 3 3 3 1 7 0 . 6 5 8 3 0 . 0 0 0 0 0 0 5 0 . 0 0 0 0 0 6 6 0 . 0 0 0 0 0 7 1 0.0 0. 1 0.1 0 . 6 4 0 - 3 5 . 3 3 0 . 0 0 0 0 2 3 5 0 . 0 0 0 1 9 3 0 0 . 0 0 0 1 8 6 4 0 . 0 0 0 3 7 9 4 1 0 . 5 2 4 . 2 3 4 . 7 0 . 8 8 7 <A> -<*• -lr -L- .JL. t<# <J- «JU J - » -JU - T T > T 1 T * -|~ T - * r - ~«" " V 6 . 0 2 4 0 . 0 0 0 0 6 8 2 0 . 0 0 0 0 3 0 3 0 . 0 0 0 0 9 8 5 3.7 8.6 1 2 . 3 0 . 8 8 5 6 . 0 2 5 0 . 0 0 0 0 0 0 0 5 0 . 0 0 5 6 1 3 . 1 2 0 2 - 1 3 . 1 2 0 2 5 0 . 0 0 5 6 0 . 0 0 0 0 1 5 7 OHMS OHMS ME T E R S * * 2 AMPS/CM MET E R * * 2 J O U L E S WATTS WATTS OHMS 1/M P FARAD/M MICROHENRY/M OHMS/.M METERS METERS J O U L E S J O U L E S J O U L E S P F A R A D MICROHENRY . OHMS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS J O U L E S J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS METERS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS METERS J O U L E S OHMS OHMS OHMS OHMS . ... ...... J O U L E S 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 . 5 9 -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 -E ( 2 , K ) . . . ET ( K) . .. I ( 1» K ) . . . 1 ( 2 » K ) . . . I ( K ) . . . 0.0 0 .0 0.0 -348.65 349. - 0 . 7 0 10425 .7 0.01 -217.85 218. - 0 . 3 0 13028.7 0.02 76. 41 76. 1.03 5 855 .8 0.02 313.34 313. 1.91 5 710.8 - 0 . 0 2 315.16 315. 1.05 12992.5 -0 .06 80.51 81 . 20. 19 13008.0 - 0 . 0 6 80. 51 8 1 . 11.36 5720.5 - 0 . 4 5 315.47 315. -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 . 6 6 -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 . 7 2 -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 . 6 5 27.32 27. 351.21 1659.1 - 3 . 2 9 35.40 36. 306.12 1227.7 - 4 . 7 7 41.71 42 . 245.80 739.2 - 6 . 0 0 45.95 46. 173.26 243.2 - 6 . 9 5 47.90 48 . 92.09 375 .7 - 7 . 5 4 47.46 48 . 6.33 381. 1 -7 .76 44. 66 4 5 . -79 .74 1356.4 -7 .60 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 . 9 6 14. 21 15. -345.85 2417.5 - 3 . 5 2 3.69 5 . -376.19 2403.6 - 1 . 8 9 -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 4 5 . -205 .36 442.0 6.59 -47.38 48 . -133.24 169.3 7. 30 -47 .97 4 9 . -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... •••ZPR«*«« • • • • 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 35. 287. 592442.4 -170.5 3 129.94 0.0 130. 416. 82339.5 -18.7 4 304.43 0.0 304. 721. 45240.0 18.1 5 133.27 0.0 133. 854. 197624.6 161.4 6 0.0 0.0 0. 854. 198099.2 -161.6 7 133.50 0.0 134. 988. 33133.0 -18.1 8 305.13 0.0 305. 1293. 26555.6 18.7 9 130.31 0.0 130. 1423. 119535.8 170.3 10 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. 1786.2 -5.0 14 0.11 0.0 0. 3533. 220.8 682920.5 15 0.0 0.0 0. 3533. 227.3 1327.8 16 0.0 0.0 0. 3533. 227.3 1327.8 17 0.02 0.0 0. 3533. 226.2 -6 2 8.6 18 0.03 0.0 0. 3533. 218.6 -251.5 19 0.05 0.0 0. 3533. 205.0 -154.6 20 0.0 0.0 0. 3533. 205.0 -28.8 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. 3540. 22.7 9.0 24 2.52 0.0 3. 3543. 116.0 20.4 25 2.11 0.0 2. 3545. 268.4 34.7 26 1 .56 0.0 2. 3546. 449.7 54. 5 27 0.96 0.0 1. 3547. 624.5 87.4 28 0.45 0.0 0. 3548. 758.4 163.3 29 0.11 0.0 0. 3548. 825.1 672.4 30 0.02 0.0 0. 3548. 811.5 -333.9 31 0.20 0.0 0. 3548. 720.1 -128.4 32 0.60 0.0 1. 3549. 568.9 -74.0 33 1.16 0.0 1. 3550. 387.7 -47.0 34 1 .75 0.0 2. 3552. 212.2 -29.5 35 2.27 0.0 2. 3554. 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. 3562. 109.0 19.8 39 2.14 0.0 2. 3564. 258.1 33.9 40 1.59 0.0 2. 3566. 437.9 53.3 41 0.99 0.0 1. 3567. 613.0 85.1 42 0.47 0.0 0. 3567. 749.0 156.9 43 0.13 0.0 0. 3567. 819.2 581 .9 44 0.02 0.0 0. 3567. 809. 8 -362.3 45 0.19 0.0 0. 3567. 722.6 -132.9 46 0.0 0.0 0. 3567. 722.6 -132.9 47 0.52 0.0 1. 3568. 586 .8 -78.6 48 0.99 0.0 1. 3569. 421.8 -51.3 49 1.50 0.0 2. 3570. 255.5 -33.7 50 1 .98 0.0 2. 3572. 116.1 -20. 5 51 2.33 0.0 2. 3575. 27.3 -9.8 52 2.50 0.0 3. 3577. 4.0 7.1 53 2.47 0.0 2. 3580. 50.0 13.1 54 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 -zs.... VPHAS F 1 0.0 0 .0 0.0 90.0000 0.0 2 0 .0053 47.85 74 47. 86 89.9936 359.9961 3 0 .0491 -170 .5079 170.51 -89.9835 359.9985 4 0.0042 -18.6886 18.69 -89.9870 0.0101 5 0.0073 18.1204 18.12 89.9770 179.9803 6 0.1318 161.3861 161.39 89,9531 179.9954 7 0.1318 -161.5797 161.58 -89.9532 0.0889 8 0.0099 -18.1330 18. 13 -89.9686 0.1137 9 0.0131 18.6758 18 .68 89.9597 180.0556 10 0.2426 170.2937 170.29 89.9183 180.0825 11 0.0307 -47.8838 47.88 -89.9633 180.1015 12 0.0140 -0.0103 0.02 -36.3040 233.7845 13 0.0148 -14.0021 14.00 -89.9395 180.1416 14 0.0139 -4.9800 4. 98 -89.8402 180.2456 15 220 .8434 0 .0714 220.84 0.0185 180.0836 16 220.8502 3 7.8101 224.06 9.7150 189.7800 17 220 .8502 37.8101 224.06 9.7150 189.7800 18 200.2400 -72.0439 212.81 -19.7881 191.3282 19 124.5143 -108.2.221 164.97 -40.9956 192.9072 20 74.3112 -98.5571 123.43 -52.9840 194.5657 21 3.9563 -28.2043 28.48 -82.0150 194.5657 22 3.2695 -14.9780 15.33 -77.6862 199.8088 23 3.0154 -3 .4263 4.56 -48.6500 229.6432 24 3.0765 7.7733 8.36 68.4074 347.4749 25 3.4789 19.7875 20.09 80.0285 359.9277 26 4.4199 34.1556 34.44 82.6266 3.5228 27 6.5155 53.7388 54.13 83.0869 5.3474 28 11.9901 85.7003 86.53 82.0355 6.5434 29 33 .6159 156.0949 159.67 77.8466 7.4647 30 329.2678 404.0674 521. 24 50.8240 8.2667 31 117.4910 -285 .5422 308 .77 -67.6344 9.0427 32 22.1775 -124.4077 126.37 -79.8923 9.8739 33 9.4727 -72 .7953 73.41 . -82.5858 10.8673 34 5.6177 -46.3297 46.67 -83.0862 12.2215 35 4.0303 -28 .9641 29.24 -82.0783 14.4391 36 3.3103 -15.6053 15.95 -78.0237 19.4222 37 3.0367 -4.0037 5 .03 -52.8203 45.4315 38 3.0823 7. 1858 7.82 66.7836 165.8132 39 3.4662 19.1256 19 .44 79.7275 179.5881 40 4.3748 33.3189 33.60 82.5198 183.3704 41 6.3914 52 .5161 52.90 83.0609 185.2558 42 11 .5915 83.4916 84. 29 82.0959 186.4793 43 31 .4775 150 .2835 153 .54 78.1701 187.4151 44 274.7024 386.7505 474.38 54.6144 188.2251 45 135 .0512 -301 .8708 330.70 -65 .8971 189.0046 46 23.6301 -128.5143 130.67 -79.5813 189.8352 47 23.6297 -128.5134 130.67 -79.5814 189.8353 48 10.3497 -77.2430 77.93 -82.3684 190.7417 49 6.1497 -50 .5585 50.93 -83.0649 191.9285 50 4.3656 -33.1115 33.40 -82.4891 193.7273 51 3.5127 -19.8905 20.20 -79.9847 197.1559 52 3.1247 -8.6956 9.24 -70.2346 207.6880 53 3.0382 1 .7147 3 .49 29.4390 308.0889 54 3.2207 12.2752 12.69 75.2984 354.6865 55 3.2207 12.2752 12.69 75 .2984 0.0 56 3.2207 12.2752 12.69 75.2984 0.0 .57 . 50.0056 0.0001 .. 50.01 0.0001 . . 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 CHARACTERISTIC IMPEDANCE = 46.00 TIP TO TIP CAPACITANCE = 6.50 VOLTAGE,CURRENT PEAKS,POWER LOSS RMS AT SHORT X=0 VOLTAGE TO CURRENT PHASE 90 HOT ARM LENGTH = 3.06780 TIP TO TIP DISTANCE = 0.1524 AVERAGE WIDTH OF SECTION = 0.8320 MAX.VOLTAGE ON RES#1 = 99998.8 MAX. VOLTAGE ON RES#2 = 100005.7 VOLTAGE PHASE SHIFT = 179.86 MAX.CURRENT IN RES#1 = 2182.1 MAX.CURRENT IN RES#2 = 2182.2 CURRENT DENSITY AT ROOT = 26.227 POWER LOSS DUE TO BEAM = 300000. POWER LOSS IN RESONATORS = 1259600. POWER LOSS IN RES#1 = 479595. POWER LOSS INRES#2 -= 480005. -•COUPLING LOOP* VOLTAGE INDUCED IN LOOP = 9291.0 LOOP DIMENSIONS ARE HIGHT = 1.75 LENGTH = 17.50 WIDTH = 3.00 THICKNESS = 0.50 LOOP POSITION = 3.50 LOOP SELF INDUCTANCE = 0.224169 CURRENT THROUGH LOOP = 271.14 POWER LOSS IN LOOP = 672.1 •TRANSMISSION LINE* CHAR. IMPEDANCE OF LINE = 54.17 OUTER DIAMETER = 11.100 INNER DIAMETER = 4.500 MAX.VOLTAGE WITHOUT CP(3) = 17399. MAX.CURRENT WITHOUT CP(3) = 321. MAX.VOLTAGE WITH CP(3) = 24268. MAX.CURRENT WITH CP(3) = 448. CURRENT THROUGH CP(2) = 270.21 CAPACITOR AFTER LOOP CP{2) = 120.000 POSITION OF CP(2) = 0.610 CURRENT THROUGH CP(3) = 303. CAPACITOR CP(3) = 120.000 POSITION OF CP(3) = 31.9200 LENGTH AFTER CP(3) = 2.4800 TOTAL LENGTH OF LINE = 34.4000 VSWR WITHOUT CP(2) = 2.36 VSWR WITHOUT CP(3) = 2.21 VSWR AFTER CP(3) = 4.29 POWER LOSS IN LINE = 5964. POWER LOSS INNER C. = 4244. POWER LOSS OUTER C. = 1720. •POWER TUBE* TUBE CURRENT = 223.69 TUBE VOLTAGE = 11318.6 CURRENT TO VOLTAGE PHASE = 0.00 CAPACITOR C P ( 6 ) AFTER LINE = 208.994 CURRENT THROUGH CP(6) = 343.3 TOTAL POWER LOSS = 1265886. MHZ 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 • R E S O N A T O R S * 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 COMPUTED Q U A L I T Y MEASURED Q U A L I T Y PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA F O R E S H O R T E N I N G C O N D U C T I V I T Y S K I N DEPTH A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l = LUMPED INDUCTANCE AT V O l LUMPED C A P A C I T A N C E AT V 0 2 = LUMPED INDUCTANCE AT V 0 2 LUMPED C A P A C I T A N C E AT 101 = LUMPED INDUCTANCE AT 101 LUMPED C A P A C I T A N C E AT 102 = LUMPED INDUCTANCE AT 102 E Q U I V A L E N T VOLTAGE PEAK 1 = E Q U I V A L E N T VOLTAGE PEAK 2 = ION O R B I T I N G FREQUENCY A C C E L E R A T I N G T I M E P E R I O D TIME CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN QMEAS = SHUNT R E S I S T A N C E 1 AT VPEAK= 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= DEE TO DEE C A P A C I T A N C E AVERAGE ENERGY IN C P ( 1 ) T 0 T = REACTANCE OF C T I P TOTAL 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 = R E S I S T A N C E 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 * 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 LOSS IN ROOT 1 POWER LOSS IN ROOT 2 MUTUAL INDUCTANCE 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 INDUCTANCE AT TUBE = C O U P L I N G C O E F F . A T TUBE 1 1 . 8 4 8 6 5 J O U L E S 1 1 . 8 4 8 6 2 J O U L E S 2 3.6 97 27 J O U L E S 1 1 . 8 5 0 3 7 J O U L E S 1 1 . 8 5 0 1 6 J O U L E S 2 3 . 7 0 0 5 2 J O U L E S 7 1 6 9 . 0 1 7 1 6 9 . 0 1 7 5 1 5 . 9 6 2 3 . 8 5 4 9 MHZ 1 2 . 9 8 7 0 2 METERS 3 . 2 4 6 7 5 METERS 1 4 5 . 1 4 1 4 3 1 0 * * 6 R A D / S E C 4 . 9 6 0 6 DEGREES 5 . 8 0 0 1 0 * * 7 MHOS 0 . 0 0 1 3 7 7 3 8 CM 0 . 0 0 0 0 3 2 7 1 1/M 4 7 3 9 . 5 9 P F A R A D 0 . 0 1 0 0 1 5 6 MICROHENRY 4 7 3 9 . 6 2 P F A R A D 0 . 0 1 0 0 1 5 5 MICROHENRY 7 6 3 0 . 2 8 P F A R A D 0 . 0 0 6 2 2 1 2 MICROHENRY 7 6 3 0 . 2 3 P F A R A D 0 . 0 0 6 2 2 1 3 MICROHENRY 1 1 4 5 7 4 . VOLTS 1 1 4 5 7 1 . VOLTS 4 . 6 2 0 0 MHZ 2 7 0 . 5 6 M ICROSEC 98 .79 M I C R O S E C 6 2 5 0 . 0 1 2 5 9 6 0 0 . WATTS 1 0 4 2 5 . 2 0 OHMS 104 1 7 . 7 4 OHMS 3 4 . 2 6 5 8 OHMS 2 6 0 . 0 0 P F A R A D 2 . 6 0 0 1 0 8 1 J O U L E S - 2 6 .50 OHMS 2 8 9 8 . 5 5 1 P FARAD / M 0 . 0 0 3 8 3 3 3 MICROHENRY / M 0 . 0 0 0 3 0 0 9 OHMS/M 0 . 1 0 1 6 METERS 0. 1 0 1 6 METERS 0 . 3 9 3 7 0 METERS 0 . 6 2 3 3 7 7 METER S**2 0 . 4 8 3 8 0 4 8 1/M - 0 . 0 0 3 1 0 9 8 OHMS 0 . 0 8 9 6 4 7 1 0 . 7 6 3 7 3 0 . 0 0 0 2 0 2 8 OHMS 1 4 5 5 6 . 3 8 WATTS 1 4 5 5 8 .39 WATTS 0 . 0 0 0 9 3 0 5 MICROHENRY 0 . 0 1 9 6 3 7 5 4 0 . 0 0 0 7 4 6 1 MICROHENRY 0 . 0 0 G 9 8 4 6 MICROHENRY 0 . 0 0 3 2 4 6 0 7 - 232 -• C O U P L I N G LOOP* LOOP R E S I S T A N C E LOOP REACTANCE LOOP AREA INPUT CURRENT D E N S I T Y LOOP S U R F A C E AVERAGE ENERGY IN LOOP LOSS DUE TO INPUT CURRENT = L O S S DUE TO C A V I T Y CURRENT = • T R A N S M I S S I O N L I N E * IMAG.COMPONENT OF ZTL A L F A IN L I N E D I S T R I B U T E D C A P A C I T Y D I S T R I B U T E D INDUCTANCE R E S I S T A N C E PER U N I T LENGTH = OUTER R A D I U S INNER R A D I U S AVERAGE E L E C T R I C ENERGY AVERAGE MAGNETIC ENERGY AVERAGE TOTAL ENERGY TL Q U A L I T Y FACTOR LUMPED C A P A C I T A N C E AT VTUBE= LUMPED INDUCTANCE AT VTUBE = S E R I E S R E S I S T A N C E OF L I N E = F I R S T S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOS S OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOS S TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 2 ) . AVERAGE ENERGY IN C P ( 2 ) SECOND S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOS S OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 3 ) I N I T . V O L T A G E M A X . P O S I T I O N = T H I R D S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOS S OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOS S TOTAL R E F L E C T I O N C O E F F I C I E N T P O S I T I O N OF VOLTAGE MAXIMUM= AVERAGE ENERGY IN C P ( 3 ) *POWER T U B E * R E S I S T A N C E TO MATCH , = R E A C T A N C E TO MATCH REACTANCE OF C P ( 3 ) R E S I S T A N C E OF TUBE AVERAGE ENERGY IN C P ( 6 ) 0 . 0 0 8 7 6 2 3 3 2.5 36 22 0 . 0 1 9 7 5 8 3 5 . 5 8 3 0 . 0 8 1 2 9 0 0 . 0 0 4 1 2 0 2 3 2 2 . 1 0 34 9.96 - 0 . 0 0 5 0 6 3 2 0 . 0 0 0 0 4 5 2 2 6 1 . 5 3 4 7 0 . 1 8 0 5 6 7 0 . 0 0 4 8 9 9 0 . 1 4 0 9 7 0 1 0 . 0 5 7 1 5 0 0 0 . 1 1 4 9 1 5 9 0 . 1 1 4 9 1 8 8 0 . 2 2 9 8 3 4 7 5 5 9 3 . 3 2 5 . 1 6 7 2 9 9 . 1 8 6 5 9 0 . 2 3 8 4 0 . 0 0 1 6 6 2 0 0 . 0 0 1 9 0 3 1 0 . 0 0 3 5 6 5 1 2 5 . 7 6 3 . 3 8 9 . 0 0 . 4 0 5 - 5 7 . 4 2 0 . 0 0 7 2 2 0 6 0 . 0 8 9 5 2 2 1 0 . 0 8 5 9 5 6 2 0 . 1 7 5 4 7 8 2 1 3 8 8 . 9 3 4 2 6 . 1 4 8 1 5 . 0 0 . 3 7 8 - 5 7 . 4 2 3 1 . 9 8 0 0 . 0 1 9 6 1 4 7 0 . 0 0 4 0 6 6 7 0 . 0 2 3 6 8 1 4 3 0 5 . 8 7 5 4 . 2 1 0 6 0 . 0 0 . 6 2 2 3 6 . 7 6 0 0 . 0 0 9 0 7 5 9 5 0 . 5 9 9 1 3 2 . 9 6 6 7 - 3 2 . 9 6 6 7 5 0 . 5 9 9 1 0 . 0 0 6 6 9 3 5 OHMS OHMS -MET ER S**2 AMPS/CM MET ER** 2 J O U L E S WATTS WATTS OHMS 1/M PFARAD/M MICROHENRY/M OHMS/M METERS METERS J O U L E S J O U L E S J O U L E S P F A R A D MICROHENRY OHMS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS J O U L E S J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS METERS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS METERS J O U L E S OHMS OHMS OHMS OHMS J O U L E S STN. 1 0.0 0.0 0.0 2 0.6136 29359.91 -3.91 3 0.6136 56151 .70 -7. 14 4 0.6136 7 8031.81 -9.08 5 0.6136 93086 .44 -9.31 6 0.6136 99998.75 -7.63 7 0.0 -100005 .50 -238.84 8 0.6136 -93092 .81 -226.38 9 0.6136 -78037 .25 -195.22 10 0.6136 -56155.87 -147.89 11 0.6136 -29362 .35 -88.29 12 0.6136 -0.55 -21.31 13 0.0 -25618.55 -79.85 14 0.0 -4316.61 -31 .27 15 0.0 -9290.97 -22.45 16 0.0 -9272.03 -8844.43 17 0.0 -9272 .03 -8844.43 18 0.2033 -9223.97 - 10244.24 19 0.2033 -9086 .71 -11545.02 20 0.2033 -8861 .56 -12734. 17 21 0.0 -8861 .56 -12734.17 22 0 .3684 -6754.96 -12438.21 23 0.3684 -4434.34 -11748.32 24 0.3684 -1973.19 - 10686.32 25 0.3684 5 50 .53 -9285.84 26 0.3684 3056.87 -7591 .23 27 0.3684 5466 .41 -5656.17 2 8 0 .3684 7702.85 -3541.90 29 0.3684 96 95 .32 -1315.39 30 0.3684 11380 .70 952.83 31 0.3684 12705 .58 3190.95 32 0.3684 13628.00 5328.06 33 0.3684 14113.71 7296.49 34 0 .3684 14162.18 9033 .90 35 0.3684 13756 .99 10485.22 36 0 .3684 12 916.00 11604.52 37 0.3684 11665 .81 12356.30 38 0.3684 10046.04 12716.78 39 0.3684 8108.00 12674.50 40 0.3684 5913.04 12230.81 41 0.3684 3530.71 11399.74 42 0.3684 103 6.50 10207.61 43 0.3684 -1490 .61 8692.16 44 0 .3684 -3970.57 6901.36 45 0.3684 -6324.80 4891.94 46 0 .3684 . -8478.70 2727.54 47 0.3684 -10364 .07 476.68 48 0.3684 -11921.15 -1739.32 49 0.3684 -13100 .63 -3998.73 50 0 .3684 -13 865.11 -6081 .56 51 0.3684 -14190.3.8 -7971.82 52 0.36.84 -14066.11 -9609.67 53 0.3684 -13496.25 -10943.21 54 0.3684 -12493.82 -11930.20 55 - 0.3684 -11105.44 -12539.38 56 0.3684 -9360.20 -12751.43 57 0.3684 -7318.39 -12559.64 58 0.3684 -5044.71 -11970.06 - 233 -ET ( K ) . .< 0.0 29359.9 56151.7 78031.8 93086 .4 99998.8 L00005 .7 93093.1 78037.4 56156.0 29362.5 21.3 25618.7 4316.7 9291.0 12813.8 12813.8 13785.0 14692.0 15514.1 15514. 1 14154.1 12557.3 10867.0 9302.1 8183.6 7866 .0 8478.1 97 84. 1 11420.5 13100.1 14632.5 15892.6 16798.2 17297.2 17363.4 16993.2 16206.1 15046.0 13585.1 11934.0 10260. 1 8819.0 7 96 2.0 7995.9 8906.6 10375.0 12054.7 13697.3 15140.2 16276.2 17035.3 17375.3 17278.6 16750.1 15818.1 14536.3 12989.7 I ( 1, 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 -271.14 -271.14 -271.14 -253.80 -234.00 -211.93 9.86 5 1.37 91.27 128.27 161.21 189.04 210.89 226.06 234.07 234.67 227.8 4 213.79 192.96 166.03 133.84 97.40 57.89 16. 53 -25.34 -66.42 -105.39 -141.03 -172. 20 -197.92 -217.37 -229.94 -235.22 -233.06 -22 3.51 -206.89 -183.71 -154.71 -120.81 -83.09 -42.73 -1.02 40.73 81. 18 I ( 2 , K ) . . . -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 -0.66 -0. 66 -0.66 -17.46 -34.10 -50.41 -204.75 -230.51 -248.97 -259.53 -261.88 -255.93 -241.87 -220.15 -191.46 -156.70 -116.97 -73.54 -27.78 18. 87 64.91 108.90 149.45 185.26 215.20 238.32 253.90 261.43 260.68 251,67 234.69 210.28 179.20 142.45 101.18 56. 70 10.43 -36.17 -81.63 -124.50 -163.43 -197.19 -224.69 -245.09 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. 0. 0.0 0.0 1 8536.86 0.0 8537. 8537. 100974.3 14.1 2 7119.47 0.0 7119. 15656. 201388.9 31.0 3 4769.78 0.0 4770. 20426. 298096.8 56.9 4 2291.92 0.0 2292. 22718. 381418.3 114.0 5 533.90 0.0 534. 23252. 430059.4 530.0 6 15000.00 0.0 15000. 38252. 430119.2 -530.0 7 533.94 0.0 534. 38786. 364345.4 -114.0 8 2292.20 0.0 2292. 41078. 233519.4 -56.9 9 4770.39 0.0 4770. 45848. 102226.0 -31.0 10 7120.43 0.0 7120. 52969. 22706.9 -14.1 11 8538.01 0.0 8538. 61507. 0.0 -0.3 12 0.0 0.0 0. 61507. 16815.1 -12.1 13 0.0 0.0 0. 1259600. 411.9 -2.0 14 322.10 0.0 322. 1259922. 34.3** *X* 1^*. «Jt*< J^U -T* " ¥ " " V * - V -lr- «V 15 0.0 0.0 0. 1259922. 65.2 68.6 16 0.0 0.0 0. 1259922. 65.2 68.6 17 34.43 0.0 34. 1259956. 75.4 77.9 18 30.03 0.0 30. 1259986. 85.7 90.3 19 25.72 0.0 26. 1260011. 95.5 106.9 20 0.0 0.0 0. 1260011. 95.5 -124.1 21 43 .99 0.0 44. 1260054. 79.5 -91.2 22 56.89 0.0 57. 1260110. 62.6 -72.5 23 69.67 0.0 70. 1260179. 46.9 -62.7 24 80.73 0.0 81. 1260259. 34.3 -64.0 25 88 .68 0.0 89. 1260347. 26.6 -102.6 26 92.52 0.0 93. 1260439. 24.5 478.4 27 91 .76 0.0 92. 1260530. 28.5 80.3 28 86.50 0.0 86. 1260616. 38.0 61.8 29 77.40 0.0 77. 1260693. 51.7 65.0 30 65.61 0.0 66. 1260758. 68.1 77.5 31 52.61 0.0 53. 1260810. 84.9 100.0 32 40.03 0.0 40. 1260850. 100.2 140.3 33 29.45 0.0 29. 1260879. 111.9 228.9 34 22.20 0.0 22. 1260901. 118.6 586.3 35 19.20 0.0 19. 1260920. 119.5 -1091.3 36 20.83 0.0 21. 1260940. 114.5 -280.9 37 26.87 0.0 27. 1260966. 104. 1 -159.1 38 36.57 0.0 37. 1261002. 89.8 -109.6 39 48.71 0.0 49. 1261050. 73.2 -83.1 40 61.76 0.0 62. 1261111. 56.5 -67.9 41 74.09 0.0 74. 1261185. 41.7 -61.5 42 84.14 0.0 84. 1261269. 30.8 -70.2 43 90.65 0.0 91. 1261359. 25. 1 -172.9 44 92.80 0.0 93. 1261451. 25.3 151.9 45 90.33 0.0 90. 1261541. 31.4 68.6 46 83.54 0.0 84. 1261624. 42.7 61.7 47 73.29 0.0 73. 1261697. 57.6 68.7 4 8 60.86 0.0 61. 1261757. 74.3 84.5 49 47.82 0.0 48. 1261804. 90.8 112.1 50 35.81 0.0 36. 1261839. 105.0 164.3 51 26 .33 0.0 26. 1261865. 115.0 296. 8 52 20.59 0.0 21 . 1261885. 119.6 1369.9 53 19.29 0.0 19. 1261904. 118.3 -528.5 54 22.61 0.0 23. 1261926. 111.2 -219.3 55 30.13 0.0 30. 1261956. 99. 1 -136.5 56 40.90 0.0 41 . 1261996. 83.7 -98.0 57 53.57 0.0 54. 1262 049. 66.8 -76.4 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 0.3684 0.3684 0.3684 0 .3684 0.3684 0.3683 0.3684 0.3683 0.3684 0.3683 0.3683 0.3684 0.3683 0 .3683 0.3684 0 .3683 0.3683 0 .3684 0.3683 0 .3684 0.3683 0.3683 0.3684 0.3683 0.3683 0.3684 0.3683 0 .3683 0.3684 0 .3683 0.3684 0.3683 0.3683 0 .3684 0.3683 0.3683 0.3684 0 .3683 0.3683 0.3684 0.3683 0.3684 0.3683 0.3683 0.3684 0.3683 0.3684 0.0 0.3100 0.3100 0.3100 0 .3100 0.3100 0.3100 0.3100 0.3100 0.0 0.0 0.0 -2611.18 -94 .86 2424.53 4867 .16 7155.66 9217.52 10987.38 12409.31 13438.10 14041 .24 14199.56 13908 .07 13175.96 12026 .46 10495.95 8632 .86 6496.30 4153 .91 1679.79 -847.52 -3348.14 -5742.63 -7955.27 -9916 .04 -11562.62 -12842 .98 -13716.56 -14155 .57 -14146.21 -13688 .66 -12797.51 -11500 .86 -9839.91 -7867 .21 -5645.15 -3244.30 -740.64 1786 .66 4257.32 6593 .15 8720.29 10571 .11 12087.18 13220 .30 13934.67 14207.61 14030.49 13408 .86 13408.82 10664 .71 7681 .13 4525 .05 1267.34 -2018.85 -5259.74 -8382.54 -11317.15 -11317.15 -11317.15 -11317.15 -11001 -9684 -8060 -6181 -4106 -1900 364 2618 4789 6808 8612 10143 11353 12204 12667 12730 12390 11657 10554 9118 7393 5433 3301 1065 -1204 -3436 -5559 -7507 -9216 -10634 -11715 -12425 -12741 -12654 -12166 -11293 • 10062 -8512 -6693 -4662 -2482 -225 2039 4240 6306 8172 9780 11078 11078 10339 9367 8185 6819 5300 3663 1942 179 179 179 179 - 235 -.37 11307.0 .20 .30 .07 .02 .89 .44 .42 .45 .93 .76 .86 .78 .11 .95 .60 .05 .11 .91 .43 .04 .55 .92 .55 .52 .52 .84 .02 .53 .23 . 13 .05 .46 .34 .43 . 19 .29 .59 .32 .01 .91 .20 .79 . 16 .31 .84 .54 .55 .59 .01 .35 .41 .71 .90 .07 .99 .24 .24 .24 .24 96 84 .7 8417.0 7867.3 8 25 0.0 94 11.5 10993.4 12682.5 14266.1 156 05 . 1 16607.4 17214.3 17392.9 17134.0 16451.2 15381.6 13989.8 12375.1 10687.7 9157.7 8 115.9 7905.8 8613.3 9973.1 11625.2 13294.8 14800.5 16023.0 16883.7 173 33.9 17349.9 16930.8 16098.7 14900.5 13412.3 11750.0 10089.5 8698 . 1 7932.5 8074.9 9066.9 10573 12258 13883 15295 16390.6 17103.0 17393.4 17393.4 14853.6 12113.9 93 5 2.9 6936 5672 6409 8604 113 18 113 18 11318.6 11318.6 ,5 3 ,6 ,8 ,6 , 6 119.07 153.19 182.45 205.94 222.91 232.81 235.35 230.43 218.21 199.08 173.65 142.71 107.26 68.40 27.38 -14.51 -55.94 -95.60 -132.23 -164.68 -191.91 -213.07 -227.48 -234.68 -234.45 -226.80 -211.96 -190.41 -162.83 -130.08 -93.23 -53.41 -11.90 29.98 70.92 109.61 144.83 175.47 200.55 219.27 231.06 235.52 232.53 222.17 204.78 180.90 151.29 116.89 -76.07 -105.78 -133.11 -157.45 -178.26 -195.07 -207.51 -215.28 -218.23 0.0 223.69 223.69 257.71 2 84. 262.18 304. :258.33 316. •246. 31 321. 226.48 318. •199.47 307. 166.15 288. •127.56 263. -84.93 234. -39.61 203. 6.97 174. 53. 32 152. 97.99 145. 139.56 155. 176.70 179. 208.25 209. 233.21 240. 250.77 268. 260.39 292. 261.77 309. 254.85 319. 239.85 321. 217.26 315. 187.79 301. 152.36 280. 112.11 253. 68.30 223. 22. 34 192. -24.34 165. -70.25 148. 113.93 147. 154.01 163. 189.21 190. 218.41 220. 240.70 251. 255.36 278. 261.93 299. 260.20 314. 250.24 321. 232.34 319. 207.08 310. 175.27 294. 137.90 270. -96.16 242. -51.37 211. -4.96 181. 41.62 157. 86.87 146. 320.42 329. 353.80 369. 379.25 402. 396.18 426. 404.21 442. 403.17 448. 393.08 444. 374.15 432. 346.83 410. 343.33 343. 0.0 224. 0.00 224. 6 6 . 5 4 0 . 0 - 236 -6 7 . 1 2 6 2 1 1 5 . 5 0 . 6 - 6 4 . 5 59 7 8 . 1 9 0 . 0 7 8 . 1 2 6 2 1 9 3 . 3 7 . 2 - 6 2 . 2 60 8 7 . 0 5 0 . 0 8 7 . 1 2 6 2 2 8 0 . 2 8 . 1 - 8 3 . 9 61 9 2 . 0 1 0 . 0 9 2 . 1 2 6 2 3 7 2 . 2 4 . 5 - 8 3 5 . 1 62 92 . 4 4 0 . 0 9 2 . 1 2 6 2 4 6 4 . 2 7 . 0 9 6 . 5 6 3 8 8 . 3 0 0 . 0 8 8 . 1 2 6 2 5 5 2 . 3 5 . 1 6 3 . 4 64 8 0 . 0 9 0 . 0 8 0 . 1 2 6 2 6 3 2 . 4 7 . 9 6 3 . 2 6 5 6 8 . 8 7 0 . 0 6 9 . 1 2 6 2 7 0 0 . 6 3 . 7 7 3 . 6 66 5 6 . 0 3 0 . 0 5 6 . 1 2 6 2 7 5 6 . 8 0 . 6 9 3 . 1 6 7 4 3 . 1 8 0 . 0 4 3 . 1 2 6 2 7 9 9 . 9 6 . 4 1 2 7 . 4 68 3 1 . 9 5 0 . 0 3 2 . 1 2 6 2 8 3 0 . 1 0 9 . 2 1 9 7 . 5 69 2 3 . 7 4 0 . 0 2 4 . 1 2 6 2 8 5 3 . 1 1 7 . 3 4 1 9 . 7 70 1 9 . 5 9 0 . 0 2 0 . 1 2 6 2 8 7 2 . 1 1 9 . 8 - 4 1 2 0 . 9 71 2 0 . 0 2 0 . 0 2 0 . 1 2 6 2 8 9 2 . 1 1 6 . 2 - 3 4 8 . 0 72 2 4 . 9 8 0 . 0 2 5 . 1 2 6 2 9 1 6 . 1 0 7 . 1 - 1 7 9 . 5 7 3 3 3 . 8 4 0 . 0 3 4 . 1 2 6 2 9 4 9 . 9 3 . 7 - 1 1 9 . 3 7 4 4 5 . 4 9 0 . 0 4 5 . 1 2 6 2 9 9 4 . 7 7 . 5 - 8 8 . 6 7 5 5 8 . 4 6 0 . 0 5 8 . 1 2 6 3 0 5 2 . 6 0 . 6 - 7 1 . 0 76 71 . 1 3 0 . 0 7 1 . 1 2 6 3 1 2 3 . 4 5 . 2 - 6 2 . 3 77 8 1 . 9 0 0 . 0 8 2 . 1 2 6 3 2 0 4 . 3 3 . 2 - 6 5 . 5 78 8 9 . 4 2 0 . 0 8 9 . 1 2 6 3 2 9 3 . 2 6 . 1 - 1 1 6 . 5 79 9 2 . 7 4 0 . 0 9 3 . 1 2 6 3 3 8 5 . 2 4 . 7 2 8 4 . 5 80 91 . 4 5 0 . 0 9 1 . 1 2 6 3 4 7 6 . 2 9 . 4 7 5 . 9 81 8 5 . 7 1 0 . 0 8 6 . 1 2 6 3 5 6 1 . 3 9 . 4 6 1 . 7 82 7 6 . 2 4 0 . 0 7 6 . 1 2 6 3 6 3 7 . 5 3 . 5 6 6 . 1 83 6 4 . 2 3 0 . 0 6 4 . 1 2 6 3 7 0 1 . 6 9 . 9 7 9 . 6 84 5 1 . 1 9 0 . 0 5 1 . 1 2 6 3 7 5 2 . 8 6 . 7 1 0 3 . 6 85 3 8 . 7 6 0 . 0 3 9 . 1 2 6 3 7 9 0 . 1 0 1 . 6 1 4 7 . 1 86 2 8 . 5 0 0 . 0 2 9 . 1 2 6 3 8 1 8 . 1 1 2 . 8 2 4 6 . 5 87 2 1 . 7 1 0 . 0 2 2 . 1 2 6 3 8 3 9 . 1 1 3 . 9 7 1 2 . 5 88 1 9 . 2 3 0 . 0 1 9 . 1 2 6 3 8 5 8 . 1 1 9 . 1 - 8 2 2 . 7 89 2 1 . 3 8 0 . 0 2 1 . 1 2 6 3 8 7 9 . 1 1 3 . 4 - 2 5 8 . 8 90 2 7 . 8 9 0 . 0 2 8 . 1 2 6 3 9 0 6 . 1 0 2 . 5 - 1 5 1 . 6 91 3 7 . 9 4 0 . 0 3 8 . 1 2 6 3 9 4 3 . 8 7 . 8 - 1 0 5 . 8 92 5 0 . 2 6 0 . 0 5 0 . 1 2 6 3 9 9 3 . " 7 1 . 2 - 8 1 . 0 9 3 6 3 . 3 2 0 . 0 6 3 . 1 2 6 4 0 5 6 . 5 4 . 6 - 6 6 . 8 94 7 5 . 4 5 0 . 0 7 5 . 1 2 6 4 1 3 1 . 4 0 . 3 - 6 1 . 6 95 8 5 . 1 6 0 . 0 8 5 . 1 2 6 4 2 1 6 . 2 9 . 9 - 7 3 . 5 96 91 . 2 0 0 . 0 9 1 . 1 2 6 4 3 0 7 . 2 4 . 9 - 2 2 7 . 2 9 7 9 2 . 8 3 0 . 0 9 3 . 1 2 6 4 3 9 9 . 2 5 . 8 1 2 8 . 0 98 8 9 . 8 3 0 . 0 9 0 . 1 2 6 4 4 8 8 . 3 2 . 5 6 6 . 7 99 8 2 . 59 0 . 0 8 3 . 1 2 6 4 5 7 0 . 4 4 . 2 6 2 . 1 100 7 2 . 0 2 o . c 7 2 . 1 2 6 4 6 4 2 . 5 9 . 4 7 0 . 2 101 5 9 . 4 4 0 . 0 5 9 . 1 2 6 4 7 0 1 . 7 6 . 2 8 7 . 1 102 4 6 . 4 4 0 . 0 4 6 . 1 2 6 4 7 4 7 . 9 2 . 5 1 1 6 . 6 103 3 4 . 6 5 0 . 0 3 5 . 1 2 6 4 7 8 1 . 1 0 6 . 2 1 7 3 . 5 104 25 . 5 4 0 . 0 2 6 . 1 2 6 4 8 0 6 . 1 1 5 . 6 3 2 6 . 5 105 2 0 . 2 8 0 . 0 2 0 . 1 2 6 4 8 2 6 . 1 1 9 . 6 2 3 2 6 . 4 106 0 . 0 0 . 0 0 . 1 2 6 4 8 2 6 . 1 1 9 . 6 - 5 8 . 9 107 9 3 . 0 5 0 . 0 9 3 . 1 2 6 4 9 1 9 . 8 7 . 2 - 4 5 . 3 108 1 1 3 . 3 6 0 . 0 1 1 3 . 1 2 6 5 0 3 2 . 5 8 . 0 - 3 5 . 3 109 1 3 0 . 7 2 0 . 0 1 3 1 . 1 2 6 5 1 6 2 . 3 4 . 6 - 2 8 . 4 110 1 4 3 . 5 8 0 . 0 1 4 4 . 1 2 6 5 3 0 5 . 1 9 . 0 - 2 7 . 8 111 1 5 0 . 7 9 0 . 0 1 5 1 . 1 2 6 5 4 5 5 . 1 2 . 7 - 1 4 6 . 2 112 1 5 1 . 7 1 0 . 0 1 5 2 . 1 2 6 5 6 0 6 . 1 6 . 2 3 1 . 4 113 1 4 6 . 2 6 0 . 0 1 4 6 . 1 2 6 5 7 5 2 . 2 9 . 2 2 7 . 2 114 1 3 4 . 9 1 0 . 0 1 3 5 . 1 2 6 5 8 8 6 . 5 0 . 6 3 3 . 0 115 0 . 0 ' 0 . 0 0 . 1 2 6 5 8 8 6 . 0 . 0 3 3 . 0 116 0 . 0 0 . 0 0 . 1 2 6 5 8 8 6 . 5 0 . 6 0 . 0 117 0 . 0 0 , 0 0 . 1 2 6 5 8 8 6 . 5 0 . 6 - 0 . 0 118 . . . . . Z S. . . . . . -VPHASF 1 0.0 0.0 0.0 90.0000 0.0 2 0 .0020 14.0705 14.07 89.9919 359.9922 3 0.0048 31 .0458 31.05 89.9911 359.9927 4 0 .0108 56.8531 56.85 89.9890 359.9932 5 0.0341 114.0342 114.03 89.98 28 359.9941 6 0.6531 529.9675 529.97 89.9293 359.9956 7 0.6531 -530.0046 530.00 -89.9293 180.1368 8 0.0357 -114.0361 114.04 -89.9820 180.1393 9 0.0138 -56.8537 56.85 -89.9860 180.1433 10 0 .00 94 -31.0462 31.05 -89.9825 180.1509 11 0.0087 -14.0708 14.07 -89.9645 • 180.1723 12 0 .0098 -0.0003 0.01 -1.6310 268.5090 13 0.0088 -12.1419 12. 14 -89.9586 180.1786 14 0 .00 95 -1.9800 1.98 -89.7246 180.4150 15 34.2659 -0.0000 34.27 -0.0000 180.1384 16 34.2746 32.5362 47.26 43.5095 223.6479 17 34.2746 32.5362 47.26 43.5095 223.6479 18 38 .9369 37.6848 54. 19 44.0638 227.9999 19 45.0662 42.7705 62. 13 43.5029 231.7948 20 53 . 1013 47.4545 71.22 41.7859 235.1664 21 59.9696 -46.1662 75.68 -37.5900 235.1664 22 45 .1838 -39.3745 5 9.93 -41.0699 241.4944 23 35.8429 -30.9503 47.36 -40.8105 249.3212 24 30 .0724 -22 .4652 37.54 -36.7611 259.5381 25 26.6529 -14.3046 30.25 -28.2224 273.3928 26 24.8990 -6.4474 25.72 -14.5174 291.9338 27 24.4800 1.2560 24.51 2.9372 314.0225 28 2 5 . 31 94 8.9898 26.87 19.5477 33 5.3062 29 27.5709 16.9319 32.35 31.5550 352.2737 30 31 .6663 25.2051 40.47 38.5184 4.7859 31 38.4429 33.7424 51. 15 41.2743 14.0980 32 49.3352 41.8935 64.72 40.3366 21.3537 33 66.3495 47.3642 81.52 35.5215 27.3297 34 90.3155 44.1490 100.53 26.0508 32.5333 35 113.9754 23.0642 116.29 11.4400 37.3137 36 118 .1323 -12 .9411 118.84 -6.2517 41.9384 37 98.1854 -40.0292 106.03 -22.1802 46.6464 38 72.9036 -47.7215 87 . 13 -33.2081 51.6918 39 53.7151 -44.0031 69.44 -39.3242 57.3925 40 41.2053 -36.2950 54.91 -41.3747 64.1983 41 33.3 75 6 -27.7604 43.41 -39.7522 72.7912 42 28.5872 -19.3863 34.54 -34.1430 84.2019 43 25.8439 -11.3539 28.23 -23.7171 99.7310 44 24.6093 -3.5764 24.87 -8.2688 119.9133 45 24.6546 4.1144 2.5.00 9.4743 142.2797 46 25.9879 11 .9039 28.58 24.6104 162.1674 47 28.8565 19.9575 35.09 34.6682 177.3666 48 33.8232 28.3506 44. 13 39.9698 188.5362 49 41.9236 36.8684 55.83 41.3290 196.9739 50 54.8418 44.4270 70.58 39.0107 203.6833 51 74.5401 47.6270 88.46 32.5764 209.3262 52 99.9766 38.7395 107.22 21.18 06 214.3400 53 118.7169 10.3664 119.17 4.9904 219.0363 54 112.64 87 -25.2132 115.44 - 12.6161 223.6666 55 88.4449 -44.8267 99.16 -26.8773 228.4704 56 64.9101 -47.1334 80.22 -35.9346 233.7194 57 48.4027 -41.3435 63.66 -40.5025 239.7710 58 37 .8673 -33.1269 50.31 -41.1799 247.1472 59 31 .3212 -24.6034 60 27.3793 -16.3593 61 25.2398 -8.4408 62 24.4938 -0.7191 63 25 .0047 6 .9849 64 26.8655 14.8533 65 30.4273 23.0296 66 36.4058 31.5171 67 46 .0620 39.8772 68 61.2986 46.3981 69 83 .6280 46.2443 70 108.8213 30.4189 71 119.6688 -3 .4780 72 104.5647 -34.9242 73 78.9972 -47.1571 74 57.9606 -45.4914 75 43.9200 -38.3915 76 35.0730 -29.9351 77 29.6199 -21 .4927 78 26.4160 -13.3811 79 24.8251 -5 .5569 80 24.5495 2.1342 81 25 .5388 9.8766 82 27.9748 17.8444 83 32 .3267 26.1458 84 39.4885 34.6726 85 50.9676 42.6555 86 68.7723 47.4937 87 93.2572 42.6626 88 115.6483 19.2946 89 116.6405 -16.8842 90 95.1337 -41.6845 91 70.33 54 -47.5814 92 52.0142 -43.1604 93 40.1510 -35.2835 94 32.7395 -26.7581 95 28.2236 -18.4335 96 25.6716 -10.4450 97 24.5895 -2.6934 98 24.7777 4.9930 99 26.2702 12.7985 100 29.3448 20.8809 101 34.6081 29.2954 102 43.1599 37.7645 103 56.7504 45.0319 104 77.2436 47.2949 105 102 .7482 36.3836 106 119.2751 6.1315 107 23.3254 -47.3859 108 18.5523 -35.6895 109 15 .6618 -25.7505 110 13.9225 -16.9549 111 12.9669 -8.8539 112 12.6171 -1.0974 113 12.8123 6.6173 114 13.5860 14.5868 115 15 .0782 23.1429 116 15.0782 23.1429 117 15 .0782 23.1429 118 50.5991 -0.0001 - 238 39.83 31.89 26.61 24.50 25 .96 30.70 38. 16 48. 15 60.93 76.88 95 .56 112.99 119.72 110.24 92. 00 73.68 58 .33 46. 11 36.60 29.61 25 .44 24.64 27.38 33. 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 30. 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 T R A P T RI TUNE 3 - 239 -DATE 0 1 - 0 5 - 7 2 1 9 : 4 4 : 0 6 IN 2 D E E S 2 3 . 1 0 0 0 4 6 . 0 0 6 . 5 0 MAIN CYCLOTRON 80 S E C T I O N S • R E S O N A T O R S * RESONANT FREQUENCY C H A R A C T E R I S T I C IMPEDANCE T I P TO T I P C A P A C I T A N C E VOLTAGE , CURRENT P E A K S , POWER. LOS S RMS AT SHORT X=0 VOLTAGE TO CURRENT P H A S E 9 0 HOT ARM L E N G T H = 3 . 0 6 7 8 0 T I P TO T I P D I S T A N C E = 0 . 1 5 2 4 AVERAGE WIDTH OF S E C T I O N = 0 . 8 3 2 0 MAX.VOLTAGE ON RES#1 = 9 9 9 9 8 . 8 MAX. VOLTAGE ON RES#2 = 1 0 0 0 0 5 . 7 VOLTAGE PHASE S H I F T = 1 7 9 . 8 6 MAX.CURRENT IN RES#1 = 2 1 8 2 . 1 MAX.CURRENT IN RES#2 = 2 1 8 2 . 2 CURRENT D E N S I T Y AT ROOT = 2 6 . 2 2 7 POWER LOSS DUE TO BEAM = 0. POWER LOSS IN RESONATORS = 9 5 9 6 0 2 . POWER LOSS IN RES#1 = 4 7 9 5 9 5 . POWER.LOSS I N R E S # 2 = 4 8 0 0 0 7 . * C O U P L I N G L O O P * VOLTAGE INDUCED IN LOOP = 9 2 9 1 . 0 LOOP D I M E N S I O N S ARE HIGHT = 1.75 LENGTH = 1 7 . 5 0 WIDTH = 3 . 0 0 T H I C K N E S S = 0 . 5 0 LOOP P O S I T I O N = 3 . 5 0 LOOP S E L F INDUCTANCE = 0 . 2 2 4 1 6 9 CURRENT THROUGH LOOP = 2 0 6 . 5 7 POWER LOS S IN LOOP = 5 3 6 . 9 '-TRANSMISSION L I N E * C HAR. IMPEDANCE OF L I N E = 5 4 . 1 7 OUTER D I A M E T E R = 1 1 . 1 0 0 INNER D I A M E T E R = 4 . 5 0 0 MAX.VOLTAGE WITHOUT C P ( 3 ) = 1 5 9 2 8 . MAX.CURRENT WITHOUT C P ( 3 ) = 2 9 4 . MAX.VOLTAGE WITH C P ( 3 ) = 2 3 2 8 5 . MAX.CURRENT WITH C P ( 3 ) = 4 3 0 . CURRENT THROUGH C P ( 2 ) = 2 2 8 . 9 9 C A P A C I T O R A F T E R LOOP C P ( 2 ) = 1 2 0 . 0 0 0 P O S I T I O N OF C P ( 2 ) = 0 . 6 1 0 CURRENT THROUGH C P ( 3 ) = 2 7 6 . C A P A C I T O R C P ( 3 ) = 1 2 0 . 0 0 0 P O S I T I O N OF C P ( 3 ) = 3 1 . 9 2 0 0 LENGTH A F T E R C P ( 3 ) = 2 . 4 8 0 0 TOTAL L E N G T H OF L I N E = 3 4 . 4 0 0 0 VSWR-WITHOUT C P ( 2 ) = 1.96 VSWR WITHOUT C P ( 3 ) = 2 . 4 4 VSWR A F T E R C P ( 3 ) = 5 . 1 9 POWER LOSS IN L I N E = 4 9 7 9 . POWER LOSS INNER C. = 3 5 4 3 . POWER LOS S OUTER C. = 1 4 3 6 . •POWER T U B E * TUBE CURRENT = 1 7 0 . 5 9 TUBE V O L T A G E = 1 1 3 1 1 . 8 CURRENT TO VOLTAGE PHASE = - 0 . 5 8 C A P A C I T O R C P ( 6 ) A F T E R L I N E = 2 0 8 . 9 9 4 CURRENT THROUGH C P ( 6 ) = 3 4 3 . 1 TOTAL POWER L O S S - - ' _ - 964769. MH Z OHMS PFARAD D E G R E E S METERS M E T E R S METERS VOLTS VOLTS D E G R E E S AMPS AMPS AMPS/CM WATTS WATTS WATTS WATTS VOLTS I N C H E S I N C H E S I N C H E S I N C H E S I N C H E S M ICROHENRY AMPS WATTS OHMS I N C H E S I N C H E S VOLTS AMPS VOLTS AMPS AMPS PFARAD M E T E R S AMPS PFARAD METERS M E T E R S METERS WATTS WATTS WATTS AMPS VOLTS DEGREES PFARAD AMPS ' WATTS AVERAGE AVERAGE AVERAGE AVERAGE AVERAGE Q U A L I T Y Q U A L I T Y Q U A L I T Y - 240 -• R E S O N A T O R S * AVERAGE MAGNETIC ENERGY 1 E L E C T R I C ENERGY 1 TOTAL ENERGY 1 MAGNETIC ENERGY 2 E L E C T R I C ENERGY 2 TOTAL ENERGY 2 COMPUTEO MEASURED PUSH PUSH PUSH PUSH FREQUENCY WAVELENGTH QUARTER WAVELENGTH OMEGA F O R E S H O R T E N I N G ...CONDUCTIVITY S K I N D E P T H ' A L F A IN RESONATOR LUMPED C A P A C I T A N C E AT V O l INDUCTANCE AT V O l C A P A C I T A N C E AT V 0 2 INDUCTANCE AT V 0 2 C A P A C I T A N C E AT 101 INDUCTANCE AT 101 C A P A C I T A N C E AT 102 INDUCTANCE AT 102 VOLTAGE PEAK 1 VOLTAGE PEAK 2 LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED LUMPED EQU I V A L E N T E Q U I V A L E N T ION O R B I T I N G FREQUENCY A C C E L E R A T I N G T I M E P E R I O D T I M E CONSTANT IN RES.2Q/0M = BUNCHES IN CYCLOTRON RESONATOR POWER WHEN QMEAS = 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= DEE TO DEE 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 TOTAL 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 = R E S I S T A N C E DEE RESONATOR GAP BEAM GAP D I S T . B E T W E E N GROUND ARMS RESONATOR AREA BETA IMAG .COMPONENT OF ZRES 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 LOSS IN ROOT 1 POWER LOSS IN ROOT 2 MUTUAL INDUCTANCE 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 INDUCTANCE AT TUBE = C O U P L I N G C O E F F . A T TUBE 1 1 . 8 4 8 6 5 1 1 . 8 4 8 6 2 2 3 . 6 9 7 2 7 1 1 . 8 5 0 3 7 1 1 . 3 5 0 1 6 2 3 . 7 0 0 5 2 7 1 6 8 . 9 9 7 1 6 8 . 9 9 7 5 1 5 . 9 6 2 3 . 8 5 4 9 1 2 . 9 8 7 0 2 3 . 2 4 6 7 5 1 4 5 . 1 4 1 4 3 4 .96 06 5 . 8 0 0 0 . 0 0 1 3 7 7 3 8 0 . 0 0 0 0 3 2 7 1 4 7 3 9 . 5 9 0 . 0 1 0 0 1 5 6 4 7 3 9 . 6 2 0 . 0 1 0 0 1 5 5 7 6 3 0 . 2 8 0 . 0 0 6 2 2 1 2 7 6 3 0 . 2 3 0 . 0 0 6 2 2 1 3 9 9 9 9 9 . 1 0 0 0 0 6 . 4.6 2 0 0 0.0 9 8 . 7 9 0.0 9 5 9 6 0 2 . 1 0 4 2 5 . 2 0 1 0 4 1 7 . 6 8 4 4 . 9 7 8 3 2 6 0 . 0 0 2 . 6 0 0 1 0 8 1 - 2 6 . 5 0 2 8 9 8 . 5 5 1 0 . 0 0 3 8 3 3 3 0 . 0 0 0 3 0 0 9 0 . 1 0 1 6 0 . 1 0 1 6 0 . 3 9 3 7 0 0 . 6 2 3 3 7 7 0 . 4 8 3 8 0 4 8 - 0 . 0 0 3 1 0 9 8 0 . 0 8 9 6 4 7 1 0 . 7 6 3 7 3 0 . 0 0 0 2 0 2 8 1 4 5 5 6 . 3 8 1 4 5 5 8 . 3 9 0 . 0 0 0 9 3 0 5 0 . 0 1 9 6 3 7 5 4 0 . 0 0 0 7 4 6 1 0 . 0 0 1 1 2 7 0 0 . 0 0 3 1 2 9 3 7 J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S J O U L E S MHZ METERS METERS 1 0 * * 6 R A D / S E C D E G R E E S 1 0 * * 7 MHOS CM 1/M P F A R A D MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY PFARAD MICROHENRY V O L T S V O L T S MHZ M I C R O S E C M I C R O S E C WATTS OHMS OHMS OHMS PF A R A D J O U L E S OHMS PFARAD/M MICROHENRY/M OHMS/M METERS METERS METERS METER S**2 1/M OHMS OHMS WATTS WATTS MICROHENRY MICROHENRY MICROHENRY - 241 -• C O U P L I N G L O O P * LOOP R E S I S T A N C E LOOP R E A C T A N C E LOOP AREA . = INPUT CURRENT D E N S I T Y LOOP S U R F A C E AVERAGE ENERGY IN LOOP L O S S DUE TO INPUT CURRENT = LOSS DUE TO C A V I T Y CURRENT = • T R A N S M I S S I O N L I N E * IMAG .COMPONENT OF Z T L A L F A IN L I N E D I S T R I B U T E D C A P A C I T Y D I S T R I B U T E D INDUCTANCE R E S I S T A N C E PER U N I T LENGTH = OUTER R A D I U S INNER R A D I U S AVERAGE E L E C T R I C ENERGY AVERAGE MAGNETIC ENERGY AVERAGE TOTAL ENERGY TL Q U A L I T Y FACTOR LUMPED C A P A C I T A N C E AT VTUBE= LUMPED INDUCTANCE AT VTUBE = S E R I E S R E S I S T A N C E OF L I N E = F I R S T S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOS S INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E A C T A N C E OF C P ( 2 ) AVERAGE ENERGY IN C P ( 2 ) SECOND S E C T I O N AVERAGE MAGNETIC ENERGY AVERAGE E L E C T R I C ENERGY A V E R A G E TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T R E ACTANCE OF C P ( 3 ) I N I T . V O L T A G E M A X . P O S I T I O N = TH I R D S E C T I O N A V E R A G E MAGNETIC ENERGY A V E R A G E E L E C T R I C ENERGY AVERAGE TOTAL ENERGY POWER LOSS OUTER CONDUCTOR = POWER LOSS INNER CONDUCTOR = POWER LOSS TOTAL R E F L E C T I O N C O E F F I C I E N T P O S I T I O N OF VOLTAGE MAXIMUM= AVERAGE ENERGY IN C P ( 3 ) •POWER T U B E * R E S I S T A N C E TO MATCH REACTANCE 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 TUBE AVERAGE ENERGY IN C P ( 6 ) 0 . 0 0 3 7 6 2 3 3 2 . 5 3 6 2 2 0 . 0 1 9 7 5 8 2 7 . 108 0 . 0 0 1 2 9 0 0 . 0 0 2 3 9 1 3 1 8 6 . 9 4 3 4 9 .96 - 0 . 0 0 5 0 6 3 2 0 . 0 0 0 0 4 5 2 2 6 1 . 5 3 4 7 0 . 1 8 0 5 6 7 0 . 0 0 4 8 9 9 0 , 1 4 0 9 7 0 1 0 . 0 5 7 1 5 0 0 0 . 0 9 4 1 6 5 6 0 . 0 9 4 2 0 1 7 0. 1 8 8 3 6 7 3 5 4 9 0 . 7 7 3 . 6 6 5 9 2 1 2 . 9 4 8 9 3 0 . 3 4 2 3 0 . 0 0 0 9 7 4 2 0 . 0 0 1 4 3 4 8 0 . 0 0 2 4 0 9 0 1 5 . 2 3 7 . 6 5 2 . 8 0 . 3 2 4 - 5 7 . 4 2 0 . 0 0 5 1 8 5 8 0 . 0 7 2 6 2 8 3 0 . 0 7 0 0 8 0 0 0 . 1 4 2 7 0 8 4 1 1 3 6 . 1 2 8 0 2 . 4 3 9 3 8 .6 0 . 4 1 8 - 5 7 . 4 2 3 8 . 1 9 0 0 . 0 1 3 2 0 8 0 0.00 32 4 1 5 0 . 0 2 1 4 4 9 4 2 8 5 .0 7 0 2 . 9 9 8 7 . 9 0.6 77 3 6 . 6 7 0 0 . 0 0 7 5 3 8 1 6 6 . 3 1 5 2 3 2 . 8 0 1 7 - 3 2 . 9 6 6 7 66 . 3 0 8 4 0 . 0 0 6 6 8 5 6 OHMS OHMS " METERS* :*2 AMPS/CM MET ER**2 J O U L E S WATTS WATTS OHMS 1/M . PFARAD / M MICROHENRY/M OHMS/M METERS M E T E R S J O U L E S J O U L E S J O U L E S P F A R A D MICROHENRY OHMS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS OHMS J O U L E S J O U L E S J O U L E S -J O U L E S WATTS WATTS WATTS OHMS METERS J O U L E S J O U L E S J O U L E S WATTS WATTS WATTS M E T E R S J O U L E S OHMS OHMS OHMS OHMS J O U L E S 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 5 7 58 .METERS., 0.0 0.6136 0.6136 0.6136 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 0.3684 0.3684 0.3684 0.3684 0.3684 0.3684 0.3684 . . E d ,K ).., 0.0 29359.91 56151.70 78031 .81 93086.44 99998 .75 -100005.50 -93092.81 -78037.25 -56155 .87 -29362.35 -0 .55 -25618.55 -4316.61 -9290.97 -9276.54 -9276.54 -9229.22 -9092.66 -8868 .16 -8868.16 -6761 .58 -4440.77 -1979.25 545.02 3052 .06 5462.45 7699 .86 9693.39 11379.88 12705.91 13629.46 14121.27 14165.77 13761.51 12921 .32 11671.77 10052 .48 8114.72 5919.86 3537.43 1042 .91 -1484.69 -3965 .32 -6320.36 -8475 .23 -10361.66 -11919.88 -13100.54 -13866.21 : -14192.64 -14069.48 -13500.62 -12504.08 -11111.43 -9366.75 -7325.33 -5051 .82 - 242 -i E(2,K) . . . E T ( K ) . . 0.0 -3.91 - 7 . 14 -9.08 -9.31 -7.63 0.0 29359.9 56151.7 78031.8 93086 .4 99998 .8 -238.84 100005.7 -226.38 93093.1 -195.22 -147.89 -88.29 -21 .31 -79.85 -31.27 -22.45 -6 743.32 -6743.32 -7809.71 -8800.61 -9706.43 -9706.43 -9479.83 -8953.01 -8142.63 -7074.34 -5781.96 -4306 .43 -2694.46 -997.09 731.92 2437.81 4066.56 5566.57 6890.36 7995.97 8848.39 9420.61 96 94.51 9661 .39 9322.28 8687.93 7778.41 6622.52 5256.84 3724.63 2074.41 358.43 -1368.96 -3 05 3.07 -4640.5 5 -6081 . 11 -7329. 14 -8345. 11 -9096.82 -9560.46 -9721.33 -9574.34 -9124.13 78037.4 56156 .0 29362.5 21.3 25618.7 43 16.7 9291.0 11468.5 11468.5 12090. 1 12654.1 13147.6 .13 147.6 11644.1 9993.8 8379.7 7095.3 6538.1 6955.8 8157.7 9744.5 11403.4 12937.6 14223.2 15178.8 15752.6 15915.9 15660.6 14999.3 13965.5 12617. 1 11043. 1 .9380.5 7848.0 6786.9 I ( 1 » K ) . . 0.0 0.01 0.05 0. 10 0. 16 0. 22 0.22 1.73 3. 10 4.21 65 84 7336 8725 103 67 11998 13451 14622 15440.6 15864.0 15371.6 15463.0 14658.3 13499.6 12055.2 10429.3 4. 5, 98 .34 5.05 5.31 -206.57 -206.57 -206.57 -193.34 -178.25 -161.43 7.63 39.27 69.67 97 .87 122.97 144. 17 160.81 172.35 178.44 178.88 173.65 162.93 147.04 126.49 101.94 74. 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 31. 14 61.98 . I ( 2 » K} . . . -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 64. 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 I(K) . . . . 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 t K ) . . - 243 -.P(2,K) • • • Z P R • • » * 0.0 0.0 0. 0. 0.0 0.0 1 8536.86 0.0 8537. 8537. 100974.3 14.1 2 7119.47 0.0 7119. 15656. 201388.9 31.0 3 4769.78 0.0 4770. 20426. 298096.8 56.9 4 2291 .92 0.0 2292. 22718. 381418.3 114.0 5 533.90 0.0 534. 23252. 430059.4 530.0 6 0.0 0.0 0. 23252. 430119.2 -530.0 7 533.94 0.0 534. 23786. 364345.4 -114.0 8 2292.20 0.0 2292. 26078. 233519.4 -56.9 9 4770.39 0.0 4770. 30848. 102226.0 -31.0 10 7120.43 0.0 7120. 37969. 22706.9 -14.1 11 8538.01 0.0 8538. 46507. 0.0 -0.3 12 0.0 0.0 0. 46507. 16815.1 -12.1 13 0.0 0.0 0. 959602. 411.9 -2.0 14 186.94 0.0 187. 959789. 45.0* si. ,u j ; si, <J. *i* ~t- 1- 1- - r - T - 15 0.0 0.0 0. 959789. 68.5 94.7 16 0.0 0.0 0. 959789. 68.5 94.7 17 20.00 0.0 20. 959809. 76.1 108.3 18 17.57 0.0 18. 959827. 83.4 127.1 19 15.30 0.0 15. 959842. 90.0 154.2 20 0.0 0.0 0. 959842. 90.0 -91.5 21 43.58 0.0 44. 959886. 70.6 -70.2 22 54.93 0.0 55. 959941. 52.0 -57 .7 23 65.09 0.0 65. 960006. 36.6 -53.6 24 72 .79 0.0 73. 960078. 26.2 -69.2 25 77.06 0.0 77. 960155. 22.3 -818.9 26 77.37 0.0 77. 960233. 25.2 76.9 27 73.67 0.0 74. 960306. 34.6 54.0 28 66 .43 0.0 66. 960373. 49.4 56.5 29 56.57 0.0 57. 960429. 67.7 67.9 30 45.32 0.0 45. 960475. 87. 1 87.6 31 34.09 0.0 34. 960509. 105.3 121.4 32 24.30 0.0 24. 960533. 119.9 190.0 33 17.17 0.0 17. 960550. 129.2 409.6 34 13.61 0.0 14. 960564. 131.9 -3295.9 35 14.06 0.0 14. 960578. 127.7 -327.0 36 18.47 0.0 18. 960596. 117. 1 -169.3 37 26.28 0.0 26. 960622. 101.5 -112.0 38 36 .50 0.0 37. 960659. 82.9 -82.3 39 47.87 0.0 48. 960707. 63 .5 -64.7 40 58.93 0.0 59. 960766. 45.8 -55.1 41 68.31 0.0 68. 960834. 32. 1 -55.5 42 74.83 0.0 75. 960909. 24.0 -95.4 43 77.67 0.0 78. 960986. 22 .6 211.6 44 76 .47 0.0 76. 961063. 28.0 62.1 45 71.37 0.0 71 . 961134. 39.6 53.6 46 63 .04 0.0 63. 961197. 55.9 59.9 47 52.50 0.0 52. 961249. 74.9 74. 1 48 41 .08 0.0 4 1 . 961291. 94. 1 97 .9 49 30.23 0.0 30. 961321 . 111 .2 140.7 50 21 .30 0.0 21. 961342. 124.0 238. 1 51 15.41 0.0 15. 961357. 130.9 705.6 52 13.32 0.0 13. 961371. 131.0 -757.2 5 3 15.27 0.0 15. 961386. 124.4 -243.9 54 21.03 0.0 21. 961407. 111.7 -142.8 55 29.88 0.0 30. 961437. 94.8 -99.1 56 40.69 0.0 4 1 . 961477. 75.6 -74. 8 5 7 52.11 0.0 52. 961529. 56.6 -60.3 58 59 0.3684 -2618.25 -8384.93 60 0 .3684 -101 .68 -7380.16 61 0.3684 2418 .15 -6141.62 62 0 .3684 4861.42 -4708.53 63 0.3684 7150 .73 -3126.27 64 0.3684 9213.55 -1444.94 65 0.3683 10984 .49 282. 15 66 0 .3684 12407.60 2000.47 67 0.3683 13437 .62 3655.44 68 0.3684 14042.02 5194.75 69 0.3683 . 14201.58 6569.55 70 0 .3683 13911.28 7736.35 71 0.3684 13180 .27 8.65 8.2 1 72 0.3683 12031.75 9305 .88 73 0.3683 10502 .08 9658.85 74 0 .3684 8639.63 9705.95 75 0.3683 6503 .54 9445.66 76 0.3683 4161 .39 8886 .22 77 0.3684 1687 .29 8045.31 78 0 .3683 -840.21 6949.61 79 0.3684 -3341.25 5633.71 80 0 .3683 -5736.36 4139.42 81 0.3683 -7949.82 2513.97 82 0.3684 -9911.57 808 .77 83 0.3683 -11559.29 -922.02 84 0.3683 -12840.86 -2623.69 85 0.3684 -13715.75 -4242.40 86 0 .3683 -14156.10 -5726.73 87 0.3683 -14148 .07 -7029.77 88 0 .3684 -13691.80 -8110.26 89 0.3683 -12801 .86 -8933.91 90 0 .3684 -11506.27 -9474.66 91 0.3683 -9846.25 -9715.36 92 0 .3683 -7874.29 -9648 .38 93 0.3684 -5652 .77 -9275.84 94 0.3683 -3252.23 -8609.52 95 0.3683 -748 .64 -7670.53 96 0 .3684 1778.82 -6488.52 97 0.3683 4249 .89 -5101.04 98 0.3683 6586 .34 -3 551.96 99 0.3684 8714.31 -1890.26 100 0.3683 10566.16 -168.71 101 0.3684 12083 .40 1558.32 102 0 .3683 13217.82 3235.99 103 0.3683 13933 .58 4811.25 104 0 .3684 14207.95 6234.23 105 0.3683 14032.26 7459.80 106 0.3684 13412.01 8449.18 107 0.0 13411 .97 8449.21 108 0.3100 10668.59 7884.79 109 0.3100 7685 .67 7143 .39 110 0.3100 4530.15 6241 .63 111 0.3100 1272 .89 5199.76 112 0.3100 -2012.96 4041 . 16 113 0.3100 -5253 .65 2791.81 114 0.3100 -8376.37 1479.77 115 0.3100 -11311 .05 134.46 116 0.0 -11311.05 134.46 117 0.0 -11311 .05 134.46 118 0.0 -11311.05 134.46 2 .1 ,8 ,9 244 -8784.2 7380.9 6600.5 6767.8 7804.3 93 26 10988 12567 13925 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 90.85 116.86 139. 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 54. 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 -257.73 273. -262.22 287. -258.40 293. -246.39 292. -226.58 283. -199.59 267. -166.28 245. -127.70 217. -85.07 187. -39.75 157. 6.83 133. 53.20 121. 97.88 127. 139.46 149. 176.63 178. 208.20 208. 233.17 237. 250.76 261. 260.41 279. 261.81 290. 254.91 294. 239.94 290. 217.37 278. 187.91 259. 152.49 235. 112.25 206. 68.45 175. 22.48 147. -24.20 126. -70.11 121. -113.81 134. -153.90 159. -189.12 189. -218.34 220. -240.65 247. -255.34 269. -261.94 284. -260.24 293. -250.30 293. -232.43 286. -207.19 272. -175.39 251. -138.03 225. -96.31 195. -51.53 164. - 5 . 11 138. 41.46 123. 86.73 124. 320.33 326. 353.72 363. 379.18 393. 396.12 414. 404.17 426. 40 3.15 430. 39 3.07 424. 374. 1,6 409. 346.86 385. -344.86 345. 0.0 171. -1.73 171. - 245 -62.70 0.0 63. 961592. 40.1 -53.7 59 71.14 0.0 71. 961663. 28.3 -61.3 60 76.37 0.0 76. 961740. 22.7 -189.6 61 77.72 0.0 78. 961817. 23.8 99.9 62 75.02 0.0 75. 961892. 31.7 55.9 63 68.63 0.0 69. 961961. 45.2 55.0 64 59.33 0.0 59. 962020. 62.8 64.3 65 48.30 0.0 48. 962068. 82.1 81.6 66 36.93 0.0 37. 962105. 100.3 110.7 67 26.64 0.0 27. 962132. 116.5 166.5 68 18.73 0.0 19. 962151. 127.2 317.0 69 14.19 0.0 14. 962165. 131.7 2502.8 70 13.59 0.0 14. 962178. 129.2 -426.8 71 17.02 0.0 17. 962195. 120.2 -193.9 72 24.03 0.0 24. 962219. 105.8 -123.1 73 33.75 0.0 34. 962253. 8 7.7 -38.5 74 44.95 0.0 45. 962298. 68.3 -68.5 75 56.23 0.0 . 56. 962354. 50.0 -56.9 76 66.17 0.0 66 . 962420. 35.1 -54.0 77 73 .52 0.0 74. 962494. 25.5 -75.0 78 77.36 0.0 77. 962571. 22.3 1582.9 79 77.20 0.0 77. 962648. 26.0 71.1 80 73.06 0.0 73. 962721. 36.1 53.8 81 65 .48 0.0 65. 962787. 51.4 57 .6 82 55.38 0.0 55. 962842. 69.8 69.8 83 44 .06 0.0 44. 962886. 89.2 90.6 84 32.93 0.0 33. 962919. 107.0 126.9 85 23.38 0.0 23. 962942. 121.1 202.9 86 16.62 0.0 17. 962959. 129.6 470.9 87 13.51 0.0 14. 962973. 131. 5 -1623.2 88 14.42 0.0 14. 962987. 126.5 -296.3 89 19.26 0.0 19. 963006. 115.3 -160.3 90 27.40 0.0 27. 963034. 99 .3 -107.8 91 37.83 0.0 38. 963071. 80.5 -79.9 92 49.23 0.0 49. 963121. 61.3 -63.4 93 60 .17 0.0 60. 963181. 44.0 -54.6 94 69.28 0.0 69. 963250. 30.8 -56.9 95 75 .41 0.0 75. 963325. 23.5 -111.6 96 77.79 0.0 78. 963403. 22.9 155.5 97 76 .13 0.0 76. 963479. 29.1 59.8 98 70.63 0.0 71. 963550. 41.3 54.0 99 61 .97 0.0 62. 963612. 57.9 61.3 100 51 .27 0.0 51 . 963663. 77 .0 76.3 101 39.84 0.0 40. 963703. 96. 1 101.7 102 29.14 0.0 29. 963732. 112.7 147.9 103 20 .51 0.0 21. 963752. 124.9 258.2 104 15.03 0.0 15. 963767. 131.0 907 .0 105 13.39• 0.0 13. 963781. 130.4 -612.1 106 0.0 0.0 0. 963781. 130.4 -52,5 107 90.35 0.0 90. 963871. 91.3 -39.9 108 108.76 0.0 109. 963980. 57. 1 -30.2 109 123 .95 0.0 124. 964104. 30.3 -23.4 110 134.57 0.0 135. 964238. 14.9 -23.5 111 139 .67 0.0 140. 964378. 10.6 97 .0 112 138.79 0.0 139. 964517. 18.3 21.8 113 132.02 0.0 132. 964649. 37. 5 25.0 114 119.95 0.0 120. 964769. 66.3 32.8 115 0.0 0.0 0. 964769. 0.0 3 2.8 116 0.0 0.0 0. 964 76 9. 66.3 0.0 117 0.0 0.0 0. 964769. 66v3 0.7 118 . .... ZS .... . . VPHA'sP 1 0 .0 0.0 0.0 90.0000 0.0 2 0.0020 14.0705 14.07 89.9919 359.9922 3 0 .0048 31.0458 31.05 89.9911 359.9927 4 0.0108 56.8531 56.85 89.9890 359.9932 5 0.0341 114.0342 114.03 89.9828 359.9941 6 0.6531 529.9675 529.97 89.9293 359.9956 7 0 .6531 -530.0046 530.00 -89.9293 180.1368 8 0.0357 -114.0361 114.04 -89.9820 180.1393 9 0.0138 -56.8537 56.85 -89.9860 180.1433 10 0.0094 -31.0462 31.05 -89.9825 180.1509 11 0 .0087 -14.0708 14.07 -89.9645 180.1723 12 0.0098 -0.0003 0.01 -1.6310 268.5090 13 0 .0088 -12.1419 12. 14 -89.9586 180.1786 14 0.0095 -1.9800 1.98 -89.7246 180.4150 15 44.9783 0 .0 44.98 0.0 180.1384 16 44.9870 32 .5362 55.52 35.8757 216.0142 17 44.9870 32.5362 55 .52 35.8757 216.0142 18 50.9446 35.8308 62. 28 35.1198 220.2377 19 58 .3031 38.2633 69.74 33.2762 224.0649 20 67.1503 39.2100 77.76 30.2813 227.5839 21 45 .7303 -45 .0172 64. 17 -44.5498 227.5839 22 35.1074 -35.3125 49.79 -45.1668 234.5013 23 28.7173 -25 .8700 38.65 -42.0141 243.6181 24 24.9470 -17.0301 30.21 -34.3194 256.3376 25 22 .9301 -8 .6833 24.52 -20.7409 2 74.4053 26 22.2436 . -0 .6046 22. 25 -1.5570 297.8276 27 22 .7511 7 .4548 23.94 18.1423 321.7488 28 24.5526 15.7448 29. 17 32.6708 340.7131 29 28 .0214 24.4969 37.22 41.1606 354.1270 30 33.9418 33 .8486 47.94 44.9212 3.6800 31 43 .7893 43 .5673 61.77 44.8544 10.8610 32 60.0854 52.1272 79.55 40.9433 16.6133 33 85 .7684 54.1310 101.42 32.2573 21.5142 34 117.4833 37.0522 123.19 17.5043 25.9387 35 131 .6468 -5 .2667 131.75 -2.2910 30.15 82 36 110.7785 -43.2446 118.92 -21.3242 34.4030 37 79.2097 -54 .7865 96 .3 1 -34.6702 38 .9080 38 55.7515 -50.5114 75.23 -42.1769 43.9615 39 41.1605 -41.4267 58.40 -45.1846 49.9726 40 32.3620 -31 .7281 45.32 -44.4333 57.5335 41 27 .0842 -22 .5105 35.22 -39.7311 67.8455 42 24.0369 -13.8792 27.76 -30.0027 82.3634 43 22 .5457 -5 .6627 23.25 -14.0991 102.6362 44 22.3058 2.3775 22.43 6.0841 127.0278 45 23.2697 10.4917 25.53 24.2694 149.4889 46 25.6301 18.9260 31.86 36.4433 166.2466 47 29.8900 27.8911 40.88 43.0188 178.0188 48 37.0486 37.4382 52.67 45.2996 186.5515 49 48.9347 47.0205 67.86 43.8571 193.1186 50 68.4399 54.0999 87.24 38.3254 198.5036 51 97.5492 50.7952 109.98 27.5066 203.1936 52 126.5370 23.4730 128.70 10.5091 207.5161 53 127.2065 -22.0101 129.10 -9.8165 211.7213 54 98.7013 -50.3185 110.79 -27.0127 216.0362 55 69.3078 -54.2335 88.00 -38.0432 220.7092 56 49.4771 -47.3414 68.48 -43.7363 226.0642 57 37.3786 -37.7856 53.15 -45.3102 232.5804 58 30.0919 -28.2225 41.26 -43.1639 241.0276 59 25.7523 -19.2369 60 23.3375 -10.7880 61 22.3313 -2 .6680 62 22 .5318 5 .3685 63 23.9785 13.5714 64 26.9656 22.1794 65 32.1509 31 .3668 66 40 .7956 41 .0437 bl 55.1262 50 .1698 68 78 .2052 54.7204 69 109.5-833 43 .9845 70 131 .3056 6 .9077 71 118.3765 -35 .8434 72 86 . 8343 -53 .8476 73 60.8432 -52.2946 74 44 .2718 -43.8661 75 34.2486 -34.1672 lb 28.2196 -24.8056 11 24.6805 -16.0410 78 22 .8291 -7.7443 79 22.2810 0.3137 80 22.9277 8 .3808 81 24.8969 16.7053 82 28 .5995 25.5123 83 34.8804 34.9141 84 45.3147 44.5923 85 62.5378 52 .7479 86 89.2785 53 .2859 87 120.4673 33 .1559 88 130 .6323 -10 .5820 89 107.0179 -45.7012 90 76.00 96 -54.6824 91 53.7305 -49.5040 92 39 .9619 -40.2596 93 31.6579 -30.6109 94 26 .6831 -21 .4767 95 23.8371 -12.9125 96 22 .4976 -4.7349 97 22.3934 3.2951 98 23 .5036 11 .4272 99 26.0510 19.9050 100 30 .5828 28.9274 101 38.1723 38.5069 102 50 .7612 47.9621 103 71.3088 54.3511 104 101 .2110 48 .9577 105 128.3452 18.5403 106 124.7004 -26.5584 107 18.1876 -45.1673 108 14.6442 -33.5031 109 12.5114 -23.6207 110 11 .2532 -14.8492 111 10.6046 -6.7179 112 10.4440 1.1376 113 10.7430 9.03 92 114 11 .5549 17.3153 115 13.0356 26.3540 116 13.0356 26.3540 117 13.0356 26.3540 118 66 .3084 0.6707 - 2 4 7 32. 14 25 .71 22.49 23. 16 27.55 34.92 44.92 57.87 74.54 95 .45 118.08 131.49 123.68 102.18 80.23 62.32 48.38 37.57 29.44 24. 11 22.28 24.41 29.98 38.32 49.35 63.58 81.81 103.97 124.95 131.06 116.37 93.64 73.06 56.73 44.04 34.25 27.11 22.99 22.63 26. 13 32.79 42. 10 54.22 69.84 89.66 112.43 129.68 127.50 48.69 36.56 26.73 18.63 12.55 10.51 14.04 20.82 29.40 29.40 29.40 66.31 -36.7596 -24.8092 -6.8 130 13.4015 29.5090 39.4375 44.2927 45.1736 42.3050 34.9806 21.8695 3.0114 -16.8459 -31.8038 -40.6789 -44.7363 -44.9318 -41.3161 -33.0217 -18.7385 0.8066 20.0790 33.8608 41.7347 45.0276 44.5396 40.1462 30.8309 15.3884 -4.6312 -23.1245 -35.7318 -42.6556 -45.2126 -44.0367 -38.8298 -28.4443 -11.8852 8.3707 25 .9284 37.3826 43.4067 45.2500 43.3759 37.3144 25.8139 8.2199 -12.0231 -68.0667 -66.3898 -62.0906 -52.8439 -32.3540 6.2165 40.0773 56.2840 63.6814 63.6814 63.6814 0.5795 252.6586 269.2104 291.4910 315.9153 336.3853 351.0869 1.4714 9.1590 15.2179 20.3016 24.8249 29.0793 33.3012 37.7199 42.6050 48.3265 55.4517 64.9064 78. 1553 96.8936 120.6714 144.1854 162.4515 175.3351 184.5605 191.5479 197.1873 202.0254 206.4214 210.6401 214.9097 219.4692 224.6166 230.7812 238.6415 249.3060 264.4255 285.3308 309.7991 331.6624 347.7612 359.0852 7.3485 13.7566 19.0497 23.6911 27.9959 32.2097 32.2099 36.4668 42.9057 54.0280 76.2446 116.4787 152.0137 169.9815 179.3189 0.0 0.0 0.0 

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