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The generation and control of 1.5 megawatts of RF power for the TRIUMF cyclotron Brackhaus, Karl Heinz 1975

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THE GENERATION AND CONTROL OF 1.5 MEGAWATTS OF EF POWES FOR THE TSIUMF CYCLOTHON by KARL HEINZ BR ACKHAUS B.A.Sc, University of B r i t i s h Columbia, 1970 M.A.Sc., University of B r i t i s h Columbia, 1972 A thesis submitted in 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 th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1975 In presenting th i s thes is in pa r t i a l fu l f 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 L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of this thes is for f i nanc i a l gain sha l l not be allowed without my writ ten pe rm i ss ion . Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date O C T ~Z^_ 7 i i ABSTRACT This thesis discusses the design of a stable, 1.5 MW, 23.075 MHz, RF system for the TRIDMF cyclotron. A physical description of t h i s EF system; the RF amplifiers, trans-mission l i n e s , and resonators, i s provided. The required c h a r a c t e r i s t i c s of th i s system are presented with emphasis on an examination of the amplitude and phase modulation con-s t r a i n t s that was carried out as part of the course of study. It i s shown that to s a t i s f y the RF system c r i t e r i a i n i t i a l l y proposed, one must reduce RF system disturbances to an absolute minimum, and then further increase system s t a b i l i t y by using amplitude and phase regulating feedback systems. A method for reducing one type of system disturbance--the use of dynamic vibration dampers f o r the resonator hot arm panels—was investigated. The analysis of dampers used for the elimination of vibrations caused by ex c i t a t i o n at a single frequency i s f a i r l y well known; t h i s thesis presents the method of analysis and design that was developed for systems excited by stochastic signals. To keep the power required by the resonators within the l i m i t s imposed by the RF amplifiers, the resonators must be kept c l o s e l y tuned to the RF driving frequency. The analy-s i s and design of a pneumatic tuning system to accomplish th i s function i s presented. The design of RF feedback regulating systems .requires a knowledge of the response of RF systems to amplitude and i i i phase modulation. Due to the stringent stability require-ments imposed on the TRIUMF RF system, i t was also consi-dered imperitive to investigate the coupling between ampli-tude and phase in the same system. To this end, the genera-lized (matrix) transfer function of a modulated carrier system was derived. This function was then approximated to give the necessary amplitude, phase, and coupling transfer functions. Because i t i s intended to "square" the RF accelerating voltage waveform by introducing third harmonic power, an-other problem is introduced. This i s , given that the phase relationship between the fundamental and third harmonic RF voltages is to be such that power is fed from the fundamen-tal RF system into the circulating ion beam and from this beam into the third harmonic RF system, will this system be stable? To answer this question the relationships between the instantaneous circulating beam currant and the RF acce-lerating voltage were derived. The RF system (complete with feedback loops) was then digitally simulated to show that the system i s , indeed, stable. The analysis of the required RF feedback loops is pre-sented in this thesis. This analysis i s followed by an i n -vestigation of the proulems of measuring the RF waveform parameters. The design of the RF measurement system i s dis-cussed and measurement results are presented. iv Table of Contents 1 Description Of System • ' 1.1 Fundamental RF Amplifier Chain ...4 1.2 Third Harmonic RF System .......8 1.3 Resonators ...........10 2 RF System Parameters .......13 2. 1 Steady State Constraints 13 2.2 Modulation Constraints .............................15 2.3 V1 Modulation Constraints .. ...18 2.4 E Modulation Constraints .......21 2.5 P3 Modulation Constraints 24 2.6 P1 Modulation Constraints .......................... 27 2.7 Combination Of Errors .............................. 31 3 Methods For Achieving Waveform S t a b i l i t y ................ 33 4 Resonator Vibration Dampers ...39 4. 1 Damper Model ................40 4.1.1 Description Of Model 40 4.1.2 Optimal Choice Of Damper Parameters ........42 4.1.3 Predicted Reduction In The Amplitude Of Vibration 44 1 Excitation At k Single Frequency 45 2 White Noise Excitation - 47 4 . 2 Measurements ......53 4.2.1 Without Dampers 54 4.2.2 Hith Dampers 58 5 Pneumatic Tuning System ..67 5.1 System Transfer Functions ..68 5.1.1 Bellows Transfer Function .........68 1 Pressure-Displacement Transfer Function ........68 2 Pneumatic Input Impedance Of The Bellows . . . . . . . 7 1 5.1.2 Transfer Function Of A Pneumatic Transmission Line ..........-.................................. . 75 5.1.3 Electropneumatic Transducer 82 5.1.4 Tuning F o i l s ...84 5.2 Frequency Response Of The Pneumatic Tuning System ..86 5.3 Tuning System Construction ...........94 6 RF Feedback S t a b i l i z a t i o n 99 6.1 Equivalent C i r c u i t Of The TL-Resonator System ......107 6.2 Envelope Response Transfer Functions ..113 6.2.1 Amplitude, Phase And Frequency Modulation 117 6.2.2 Calculation Of Modulation Transfer Functions ..122 1 P a r a l l e l Resonant C i r c u i t ..............122 2 2 Coupled, Resonant C i r c u i t s .........125 6.3 The RF--Beam Interaction 132 6.3.1 Instantaneous C i r c u l a t i n g Current (Theory) ....133 6.3.2 Beam Current Calculations 139 1 Rectangular Current Pulses 139 2 Current Pulses Shaped By Resonator Voltage .....142 6.4 Frequency Response Calculations 149 6.4.1 Fundamental RF System 149 1 Fundamental Amplitude Regulation ...............153 vi 2 Fundamental Phase Regulation ........154 6.4.2 Third Harmonic RF System ....158 6.5 Simulation Of The RF Feedback System .... ........162 6.5.1 Description Of Method ......................... 162 6.5.2 Simulation Results .....166 1 No Feedback 166 2 With Feedback 171 7 RF Probes .. 180 7.1 Phase Probes ................181 7.2 Voltage Probes ... ......183 7.3 RF Measurements ...............193 8 Summary .202 A SELECTED BIBLIOGRAPHY .208 Appendix "A" .. ......210 Appendix »B" 218 v i i Table of Figures Fundamental BF Amplifier Chain ...5 Third Harmonic RF Amplifier ... •? TBIUHF Resonators ...12 An Upper Resonator Segment 13 RF Waveforms ..............15 Curves Of Constant m, P1=0°, Pm=0° .....20 Curves Of Constant m, P1=0°, Pm=90° .. ...20 Maximum m/DEL For DEL=2/10s, Vo Modulated .22 Log 10 (DEL) Versus RF Fund, phase 23 P3 Modulated, P1=0° 27 P3 Modulated, Pl=-2° ........... 28 P1 Modulated, Pm=90°, P0=0° ............ 30 P1 Modulated, Pm=90°, P0=-2° ...31 RF Control System 36 Main System Plus A Dynamic Damper 41 Amplitude Of Hot arm Vibration 46 Plot Of FrO Versus Log10(fim) .....47 Amplitude Of Vibration Squared: Hot Arms 49 Hv: Approximate And Exact, Fund. Mode ,.,........,50 Sensitivity Of Rv With Respect To M2,K2,C2 53 Frequency Response Measurement .54 Frequency Spectrum Measurement 55 Frequency Besponse—Single Hot Arm 56 Frequency Response—4 Hot Arms, Tips Connected ............ 57 v i i i Vibrational Spectrum: Single Hot Arm ...................... 59 Vibrational Spectrum: 3 Hot Arms, 16 Gpm .60 Vibrational Spectrum: 3 Hot Arms, 5 Gpm ......61 Drawing Of Damper For Fundamental Mode .................... 62 Vibrational Spectrum: 4.7 Hz Damper ....................... 64 Vibrational Spectrum: 4.7 Hz Plus 10 Hz Dampers .....65 Hot Arm Vibration With/Hithout Dampers ........66 Block Diagram Of The Resonator Tuning System .......67 Schematic Of Pneumatically Actuated Tuning F o i l s .......... 69 Tuning Bellows Model 70 Frequency Response Of Bellows «-•» ,.......,...,72 Magnitude Of Bellows Input Impedance ,.,,......76 Phase Of Bellows Input Impedance 77 Mag. of Bellows Input Impedance (Inductive) ............... 78 Phase Of Bellows Input Impedance (Inductive) ...79 A Section Of Pneumatic Transmission Line .................. 82 Freq. Response Of The Fisher E-p Transducer .............. 83 Organization Of Pneumatic System Components .....87 Computer Flow Chart .............. ....... ........... 88 PNEUMATIC System Magnitude Response 90 PNEUMATIC System Phase Response ....91 PNEUMATIC Main Line Input Conductance ,, ..........92 Schematic Of Tuning System 96 (TUNING SYST) Magnitude Of Response, Integral Control 97 (TUNING SIST) Phase Of Response, Integral Control ....98 RF Feedback Loops ........99 Resonator Waveforms (Maximum Energy Resolution) .103 ix Plots Of Constant P31 Vs V1 and £ (Vp=100 KV) .... .....106 Equivalent C i r c u i t For TL-Sesonator System ......108 TL-Resonator System (»ith Beam Current) .113 Equivalent Amplitude Transfer Function ....................114 I l l u s t r a t i o n Of Amplitude Vectors ......116 Output Amplitude Vector ,. .......119 P a r a l l e l Resonant C i r c u i t ..123 Equivalent C i r c u i t For The TL-Resonator System ............125 RF Voltage G Injected Current Waveforms ...134 Integral Approximation For Ci r c u l a t i n g Beam Current .......136 Transit Time From The Integrated Resonator Voltage ........138 Cir c u l a t i n g Beam Current Waveforms ,143 Beam Current Components ...145 Fundamental RF System Step Response ...150 Step Response (2nd Order System) ,., ,..,...,...152 Block Diagram Of Amp. Control System ....154 Bode Plots For The Fund Amp Control System ,,..,,,....,155 Amp Response (Fund Phase Control System) 156 Phase Response (Fund Phase Control System) ,.,.,,157 Third Harmonic F-B Loop (Amp Control) ,. ........158 Bode Plots For The Third Harmonic Control System 161 Block Diagram For RF-beam Interaction .167 Beam Turn-on, Transient 6 Steady State Results ............ 169 Response To Step In Fundamental Voltage Source 172 Response To Step In 3rd Harmonic Voltage .173 Response To Step In Input RF Phase 174 Response Of The 4 Uncoupled RF F-B Loops .....175 X Schematic Of System With Detector Coupling .....176 Detector Induced Coupling, Step In VR1 .177 Detector Induced Coupling, Step In VR3 ............178 Resonator Section With Pick-Up Loop ..................181 Drawing Of Resonator Tip Voltage Probe ...............185 Voltage Probe—Equivalent Circuit ....................186 Center Region Voltage Probes ... , .....188 Signal Averaging Circuit ...190 Noise Spectrum, Fund Mode, No Feedback .......,.,195 Noise Spectrum, Fund Mode, No Feedback ...196 Noise Spectrum, Fund Mode, No Feedback .................... 197 Noise Spectrum, Fund Mode, With F-B 198 Noise Spectrum, Fund Mode, With F-B ,.199 Noise Spectrum, Fund Mode, With F-B 200 i i L i s t of Tables Parameters—Maximum Current Density Mode .14 Parameters—Maximum Energy Resolution Mode 14 Optimum Hot Arm Damper Parameters ..43 Some Values Of Bp, Lp, Cp ..........81 Parameters For F i g . 5.13 To Fig. 5.15 ....89 PNEUMATIC System Phase Cross-Over Frequencies 93 Pneumatic System Damping Ratios ....93 Maximum Input Conductances 94 TL-Sesonator Parameters (23.1 MHz.) ...........110 Component Values For Figure 6.4 (23.1 MHz.) ...............110 TL-Resonator Parameters ( 6 9.3 MHz.) 110 Component Values For Figure 6.4 ( 6 9.3 MHz.) 111 Time Constants For The TL-Resonator System 131 Input Beam Current Parameters 139 Cir c u l a t i n g Beam Current (Fourier Components) .......144 CSMP Simulation Function Blocks ......164 Summary—Permissible Levels Of RF Modulation 204 ACKNOWLEDGMENT I am much indepted to my Research Supervisor, Dr. K. L. Erdman, for his guidance in my studies and in the preparation of this thesis and to Dr. W. Joho for acting as External Examiner. Thanks are also due to the other members of the committee in charge of my course of studies--Dr. A.D. Moore, Dr. R.H. Johnston and especially. Dr. J.B. Warren—for the time and effort spent on my behalf. The financial support from TRIOMF during the course of this work i s also gratefully acknowledged. 1 CHAPTER 1. DESCRIPTION OF SYSTEM TRIUMF (Tri-University Meson Facility) i s a project which was conceived 1 to provide researchers with intense beams of mesons for investigations in nuclear physics, ra-diochetnistry and radiobiology. These mesons are to be pro-duced at TRIUMF by an isochronous, sector-focused cyclotron which uses H- ions as the accelerated particle. Because the second electron of these particles has a binding energy2 of only .755 eV, these particles are not very stable; their excessive dissociation i s avoided by using low magnetic f i e l d s 3 (5.76 kG at 308 i n . , which is the 500 MeV orbit) and a very good (10 - 7 Torr) vacuum in the accelerating chamber of the cyclotron. The H_ ions are generated by a hot f i l a -ment Penning arc (Ehler's type) and accelerated to an energy of 300 keV before being injected* into the median plane of the cyclotron by means of a centrally located, spiral elec-* E.W. Vogt and J.J. Burgerjon, TRIUMF Proposal and Cost Estimate, University of British Columbia, 1966. 2 G.M. Stinson et a l . , Electric Dissociation of H~ ions hi Magnetic Fields, TRI-69-1, 1969. 3 M.K. Craddock and J.R. Richardson, Magnetic Field Tolerances for a Six-Sector 500 MeV Sr Cyclotron, TRI-67-2, U.B.C., 1968. * L.w. Root, Ph.D. Thesis (Physics), University of British Columbia, 1974. 2 trostatic inflector. The injected beam i s focussed in the median plane by an azimuthally varying magnetic f i e l d provi-ded by a magnet having six spiral segments. To keep the ion beam isochronous (that i s , keep a constant ion rotation fre-quency of 4.615 MHz.) the r e l a t i v i s t i c effect of increasing particle mass i s counteracted by a radially increasing, axial, magnetic f i e l d . The H~ ions are accelerated by an RF potential dif-ference of 200 kV maintained between the two Dees of the cyclotron. Each Dee consists of a resonant, coaxial cavity operating in a 0y4 mode at a frequency of 23.075 MHz. (the fundamental RF frequency). This frequency was chosen to be the f i f t h harmonic of the ion rotation frequency to f a c i l i -tate the placement of these resonant cavities (resonators) inside the vacuum tank. Since the resonators are operated in the push-pull mode, the peak resonator voltage of 100 kV gives an accelerating voltage of 200 kV. With the resulting 400 keV of energy gain per orbit, 1250 turns are required to reach 500 MeV. A significant increase in the duty cycle and energy resolution of the cyclotron i s obtained by combining two RF signals, that i s , by adding 11% of third harmonic1.2 to the RF waveform, thereby "flat-topping" i t . This will 1 M.K. Craddock, Effects of the Third Harmonic on the RF laveform, TEI-DN-72-15, June 1972. 2 K.L. Erdman, A. Prochazka, M. Zach, RF Sguare Wave, TRI-DN-70-33, March 1970. 3 require a total power output (absorbed by beam and copper losses) of about 1.6 MH by the fundamental EF amplifier and about 100 kW by the third harmonic RF amplifier. Extraction of the beam i s accomplished by passing the beam through a thin f o i l which strips away the H~ ions* two electrons and thereby reverses the curvature of the beam's orbit. The efficiency of this method approaches 100%. Also, by altering the position of this f o i l , the energy of the extracted beam may be varied from 200 MeV to 500 MeV. Four beams may be extracted from the machine at the same time, each having a unique energy and intensity. It is desired that the TRIUMF cyclotron should even-tually have the ability to produce a beam with an unusually high energy resolution 1 (2 parts in 10s) corresponding to an energy variation of only ±10 keV at an energy level of 500 MeV. This requirement can only be met by separated turn acceleration 2 which demands a Dee voltage of extremely high amplitude and phase stability, and a highly stable and pre-cise magnetic f i e l d . (By separated turn acceleration i t i s meant that the particle orbits in the machine do not overlap.) The required RF voltage stability of less than 2 parts in 10s for low frequency modulation i s of the same 1 J.R. Richardson, Energy. Resolution in a 500 MeV H~ Cyclotron, U.B.C., 1969. 2 M.K. Craddock and J.R. Richardson, Magnetic Field Tolerances for a Six-Sector 500 MeV H~ Cyclotron, U.B.C., 1968. 4 order as the energy resolution and i s much loser than that of any other high power RF system presently in existence. Attaining the required degree of RF sta b i l i t y is d i f f i c u l t , involving f i r s t a careful amplifier design to reduce to a minimum a l l sources of noise and then the use of feedback to further lower the noise in the RF system. 1.1 Fundamental EF Amplifier Chain This RF amplifier system1*2*3 supplies the 1.65 MW of RF power that i s required at a frequency of 23.075 MHz (the fundamental frequency of the RF accelerating waveform). Two distinct modes of operation exist for this system: (1) a self-oscillatory mode, in which the drive for the amplifier is derived from the resonators. (2) a driven mode, in which the signal i s provided by a very stable freguency generator. The f i r s t mode of operation i s to be used during start-up 1 J.B warren, TRIUMF , March 1971, IEEE Trans. Nucl. Sci., NS-18, #3,272 (1971) . 2 K.L. Erdman, R. Poirier, O.K. Fredricksson, J.F. Weldon, W.A. Grundman, TRIUMF Amplifier and Resonator System, 6th Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 451. 3 Designed and manufactured by Continental Electronics Mfg. Co. Of Dallas, Texas. 5 when thermal expansion of the resonators causes t h e i r reson 120 kw I PA ' •CW100 .000E 1600W 20W FREQ. SYNTH. 5CX1500A TRANSISTOR STAGES SCREEN -fc SCREEN MOD. SUPPLY 30 kW, 30 kW (PA #k) (PA #3) (COMBINER #2) POWER f > - b l V I D E R AND PHASER 30 kW PA #2 1)50J<W (2) * tCW250,000A 1 900 kW 30. k«? |5| (2]_ 1^ 250^ 00 OA J C l a s s ' C G r o u n d e d G r i d P o w e r A m p l i f i e r s 20 kV 2.6 MW POWER S U P P L Y CROWBAR SAFETY C I R C U I T F i g . 1 . 1 . Fundamental RF Amplifier Chain ant frequency to vary with time. When a stable operating frequency has been achieved, a switch i s made to the driven mode. The frequency generator, a Schoiaandel Type ND 30 M RF 6 frequency synthesizer, i s the primary signal source of the RF system. This device was chosen because i t satisfies the stringent tolerances on the frequency modulation of the RF signal. A frequency stability better than 5x10 - 9/°C for ambient temperatures ranging from 10°C to 30°C and a long term d r i f t due to aging of less than 5 parts in 108 per month has been quoted for this device. The output of the synthesizer into a 50 ohm load i s .5 Vrms. As indicated in Figure 1.1, the output from the synthe-sizer i s amplified by a transistor and then a tube stage to the level of 1600 watts required to drive the I n i t i a l Power Amplifier (IPA) to an output of 120 -kS. The IPA consists of a single, low noise tetrode (an Eimac 4CW100,000E) in a grounded cathode configuration. This tube has an inherently low noise level (about -82 dB), but to produce an output with an even lower noise level, i t is driven to saturation. The output from the IPA is fed to the f i n a l Power Amplifiers (PA's) by means of a divider c i r c u i t . This d i v i -der consists of a combination of Pi networks which permit the proper driving of several combinations of PA's (2 PA's, 3 PA's, and 4 PA's). i The f i n a l stage of this RF amplifier chain consists of 4 PA's, each u t i l i z i n g 2 tubes in a push-pull configuration. This total of 8 tubes i s more than enough to satisfy the TRIUMF RF power requirements and, in addition, this arrange-ment offers considerable versatility of operation. The tube used i s an Eimac 4C8250,000A which exhibits current modula-7 tion noise in the range of -70 dB to -82 dB (overal noise level desired: 2/105 = -93.98 dB). The tubes are operated in a grounded grid configuration which, although producing lower gain than other configurations, features reduced feed-back capacity, reduced bias voltage and reduced screen dis-sipation. The drive power i s coupled into each PA (input impedance approximately 15 ohms) via a toroid coupling tran-sformer. The anode power supply for the PA's will deliver up to 130 amps of current at ,a level of 20 kV. In order to reduce the ripple in the DC output, the power transformer i s con-nected partly in a delta and partly in a wye configuration, giving a 12 phase output. An electronic crowbar ci r c u i t i s used to protect the tubes in the event of a malfunction. The output from the PA's i s summed by means of the 3 combiners shown in figure 1.1. The arms of these combiners are composed of 90° phase shifting networks, three of which consist of leading and one of lagging elements. The RF power i s fed in at 2 ports of these units (in phase) and at the other 2 ports the combined RF i s thus, respectively, in phase and 180° out of phase. A waster load is placed at the latter port to absorb any power appearing there (normally zero). The output i s the RF that has been combined in phase. The output from the RF amplifier chain i s fed to the resonators via a coaxial transmission line composed of two 8 l sections: (1) a (matched) section with a low voltage standing wave r a t i o or VSWR, (2) a (resonant) section with a high VSWB. The f i r s t section i s composed of about 160 feet of a commer-c i a l 50 ohm coaxial l i n e that i s 9 inches i n diameter. The resonant section of the l i n e (1.5 wavelengths long) has an outer conductor that i s 11 inches in diameter and i s f a b r i -cated from 1/4 inch thick aluminum sheet r o l l e d into a pipe. I t s center conductor i s fabricated from aluminum Unitrace pipe and i s water cooled. The purpose of th i s section of resonant l i n e i s to match the resonator coupling impedance to that of the 50 ohm l i n e . To provide protection i n case of a spark-over, spark gaps are placed at the nodes in th i s l i n e . Three Jennings vacuum capacitors (placed at each end and i n the middle) are used to "tune" t h i s resonant section. 1.2 Third Harmonic RF System This system i s to supply the 35 kffl of RF power which i s required (for 200 kW beam power) at the t h i r d harmonic of the RF fundamental frequency, that i s , at 69.225 MHz. This amplifier d i f f e r s from the fundamental RF amplifier i n the Notes TBI-D l i -on the TBI0MF Resonator 71-10, March 1971. 9 type of resonant elements used in the PA: the fundamental SF amplifier u t i l i z e s lumped elements, while the t h i r d harmonic amplifier u t i l i z e s resonant c a v i t i e s {see Figure 1.2). As 1/4 wavelength Coaxial Grid Cavity 2 kW + 15 kV O ~ 1/4 wavelength ^ • 4CX 3000 A Coaxial Plate Cavity 4CW100,000 E 1 To Resonators 150 kW T100 W Transistor Amplifier Frequency T r i p l e r Fundamental Frequency Synthesizer! Fig. 1.2. Third Harmonic RF Amplifier shown in th i s Figure, quarter wave, coaxial c a v i t i e s are used. These are tuned by means of motor driven shorting plates and the power tube, an Eimac 4CW100,00£, i s i n s t a l l e d inside the center conductor of these c a v i t i e s . A grounded cathode configuration i s used. Power i s fed into the g r i d cavity, and taken out of the plate cavity, by means of coup-l i n g loops. 1 0 The input signal for the t h i r d harmonic system i s pro-vided by a frequency t r i p l e r which takes as i t s input the drive signal for the fundamental amplifier system. The output from the t r i p l e r i s boosted by a tran s i s t o r amplifier to a l e v e l of 1 0 0 watts and then by a tetrode (a 4 C X 3 0 0 0 A ) to the l e v e l of 2 k W required to drive the coaxial grid cavity of the 4 C W 1 0 0 , 0 0 0 E . The output power from the t h i r d harmonic amplifier i s transmitted to the resonators by means of a commercial (6 1 / 8 i n . dia.) coaxial l i n e . This l i n e i s similar to that of the fundamental RF system i n that i t consists of a matched, 5 0 ohm, section and a resonant section which i s approximately 1 . 5 wavelengths long and i s used to match the coupling impedance of the resonators to that of the 5 0 ohm l i n e . The tuning of t h i s resonant (high VSWR) section i s accomplished by means of three inductive tuning stubs placed at each end, and at the middle, of the section. The center conductor of the resonant section, as well as the tuning stubs, are water cooled. 1 . 3 Resonators Figure 1 . 3 i l l u s t r a t e s schematically the nature of the TRIUHF RF resonators. These coaxial c a v i t i e s (characterist-i c impedance 4 6 ohms per segment) are excited in a push-pull mode with a peak voltage of 1 0 0 kV at the t i p of each reson-ator. Excitation i s maintained at two frequencies simul-11 taneously: at the RF fundamental frequency, and at i t s third harmonic. Due to the tip-to-tip capacitance of about 7 pF, the resonator's electrical length i s slightly less than a quarter wavelength for the fundamental frequency and three quarter wavelengths for the third harmonic. The Q of the resonators i s about 7000 and 6000, respectively, for the fundamental and third harmonic modes. A single coupling loop i s used at each frequency to feed power into the cav-it y . As indicated in Figure 1.3, the resonator as a whole i s 53 feet long and 21 feet wide. This structure i s composed of 80 resonator segments (see Figure 1.4) ) and 4 flux guides. Each segment i s 193 inches long, 32 inches wide, and weighs approximately 600 pounds. Except for the level-ing arms, which are stainless steel, the segments are made almost entirely of aluminum. The RF surfaces of the seg-ments are fabricated from "roll-bond" (copper on aluminum) which permits the incorporation of cooling channels to remove the 12 kW of heat generated by each segment. The design includes a flexible ground arm t i p for adjusting the resonant frequency of the cavity, as well as two tuning f o i l s located in pockets in the root of each segment. These f o i l s are part of a pneumatically actuated feedback tuning system. The design of the segments i s such that remote handling i s possible. R . F S K I N L O S S E S « 1 . 2 M W T O T A L R . F P O W E R ' I . 5 M W R . F C O U P L I N G L O O P S F L U X C O U P L I N G S - H O O k V + IOO kV — 2 0 0 k V Fig. 1.3. TRIUMF Resonators 13 Fig. 1.4. An Upper Resonator Segment CHAPTER 2 . RF SYSTEM PARAMETERS 2.1 Steady State Constraints Two basically different modes of operation are planned for the TRIUMF cyclotron: a maximum current density mode and a maximum energy resolution mode. Although in each case the RF voltage (Vr) may be described by Vr=Vl[cos (P1)-E*cos (3*P1 + P3) ] (2.1) 14 V1 = 102.5 kV ± 2/10* E = .24 P3 = -24° f1 = 23.055 MHz ± 1.25/106 average beam current = 100x10~6 amps maximum beam current = 400x10~6 amps Table 2.1. Parameters—Maximum Current Density Mode V1 = 112.5 kV ± 2,5/10* E = .11111 = 1/9 P3 = 0° f1 = 23.055 MHz ± 7.5/103 average beam current = 20x10-6 amps Table 2.2. Parameters—Maximum Energy Resolution Mode these modes differ in the fraction (E) and phase (P3) of third harmonic voltage which i s required 1. (P1 is the phase of an ion with respect to the peak of the fundamental RF voltage.) Tables 2.1 and 2.2 l i s t the RF waveform parameters for these modes, together with the steady state errors per-mitted. Figure 2.1 shows the RF accelerating voltage wave-forms corresponding to the two modes of cyclotron operation. 1 M.K. Craddock, Effects of the Third Harmonic on the RF Waveform, TRI-DN-72-15, June 1972. 15 -90.0 -45.0 , 0.0 45.0 90.0 P H A S E (DEG) F i g . 2.1. RF Waveforms 2.2 Modulation Constraints In addition to the permitted s t a t i c deviations l i s t e d i n the l a s t section, an estimate must be made of the con-s t r a i n t s on the time varying errors in the RF parameters. 1 6 This information i s needed to define the requirements of the feedback control systems which stabilize the RF amplitude and phase. As i s to be expected, the more severe requirements, in terms of RF sta b i l i t y , are posed by the maximum energy reso-lution mode of operation. Thus, this i s the mode for which energy resolution calculations sere performed. The energy resolution i s determined by (1) the amplitude of the beam's radial oscillations (2) the energy gain per turn (3) the phase spread of the circulating ions with res-pect to the RF fundamental voltage. It was assumed that variations in the RF accelerating vol-tage (Vr) were solely responsible for variations in the energy resolution (DEL). Single turn extraction i s , of course, required. If this is not possible, that i s , i f the ions' radial oscillations are sufficiently large to cause an overlapping of orbits, then particles from different orbits will be extracted at the same time. These particles may differ in energy by as much as 400 keV, a difference in energy much larger than the 10 keV permitted at the 500 Me? orbit i f an energy resolution of 2/10s i s to be observed. It was also assumed that the cyclotron was perfectly isoch-i K.L. Erdman, RF Stability Criteria, TRI-DI-71-2, January 1971. 17 ronous. The res u l t s obtained are therefore the minimum con-diti o n s which must be s a t i s f i e d by the TRIUMF RF system. To estimate the e f f e c t of time varying changes, the parameters of the RF waveform (Eguation 2.1) were taken to be sinusoi d a l l y modulated. The re s u l t i n g constraints are therefore expressed in terms of a permissible amplitude of modulation as a function of the frequency of modulation. To calculate the energy resolution, one must f i r s t c a l -culate the t o t a l energy gain (Et) for a given RF voltage: N E t = 2 g * y V r (t) (eV) (2.2) where (Vr) i s the resonator voltage and N i s the t o t a l number of accelerating impulses. The l a s t equation may then be approximated by the i n t e g r a l Et= (2q/ A t ) * \ Vr ( t ) d t (2.3) J 0 where ( t f ) i s the t o t a l acceleration time and . A t (=1.083x10-7 sec.) i s the time between impulses. _ T h i s ap-proximation i s based on the fact that the area given by r - r N V A t * V r ( n * A t ) i s closely approximated by the integral n£n = o \ Vr(t)dt where t f = N * A t . 18 Consideration of Equation 2.1 shows that there are four quantities which may vary with time: V1, E, P3 and P1. The constraints on each will be considered in turn. 2.3 VJ[ Modu la t i on Constraints To find the restraints on the modulation of the funda-mental RF voltage amplitude (V1) i t was assumed that V1 varied according to V1 (t) =V1 (1+m*cos (wm*t + Pm) ) (2.4) where m = modulation ratio = [max. V1 (t) - ¥1 ]/V1 wm= frequency of modulation (rad./sec.) Pm= phase of modulation at the instant a beam pulse i s injected into the cyclotron. Substituting this last equation in Equation 2.1 and then performing the integral shown in Equation 2.3 gives the re-sult: Et ._ E r e f cos(Pl)-Ecos(3Pl) 1+. m (sin:(wmtf+Pm)-sin(Pm)) 1-E J[ wmtf (2.5) where Eref (=2*q*V1*N* (1-E)) is the energy gain at P1=0 when there is no modulation present. The fractional change in energy gain with respect to Eref was defined as the energy resolution (DEL) : 1 9 nKT.= | - E r p f | J r e f (2.6) Since the t o t a l energy gain (Et) depends on the phase of the modulating signal as well as i t s amplitude, a serie s of plots were made for the energy resolution (DEL) as a func-tion of the modulating frequency f o r constant amplitudes of modulation. Figure 2.2 gives a set of constant amplitude curves with a modulation phase (Pm) of zero degrees. Figure 2.3 shows the same curves but with the phase (Pm) equal to 90 degrees. Using Figures 2.2 and 2.3, a plot was was made (Figure 2.4) of the maximum permissible modulation r a t i o (m) for an energy resolution of 2/10 5. It should be noted that, at a frequency of 10 kHz., t h i s maximum modulation r a t i o has been increased to 2/10*. For beam phases other than P1=0, the above amplitudes of modulation must be reduced. Figure 2.5 shows a plot of the energy resolution (DEL) versus the EF fundamental phase (P1) when there i s no modulation pres-ent. The RF control system b u i l t to control the amplitude of the RF voltage waveform does not control V1 d i r e c t l y ; r a-ther, i t controls the peak RF voltage. The preceeding plots were made with P3 set equal to zero. In t h i s case the peak of the waveform occurs at P1=0 and thus the results derived apply to the fundamental RF amplitude control system. 0.0 2-0 -4J) 6.0 8.0 , 10. FREQUENCY: I KHZ.) Fig. 2.2. Curves Of Constant m , P 1=0°, Pm=0° 21 a FREQUENCY I K H ZJ Fig. 2.3. Curves Of Constant m, P1=0°, Pm-90° 2.4 E Modulation. Constraints For t h i s case i t was assumed that the time dependent' variable was the f r a c t i o n of t h i r d harmonic ( E ) : E (t) = E * (1+m*cos (wm*t + Pra) ) (2.7) 22 Fig. 2.4. Maximum m/DEL For DEL=2/105, Vo Modulated 23 a 0 . 0 2 . 0 4 . 0 . 6 . 0 8 . 0 PHASE (DEG) Fig. 2.5. Log 10 (DEL) Versus RF Fund, phase 1 0 . 0 where m = modulation r a t i o wm= frequency of modulation (rad./sec.) Pm= phase of modulation at the moment of p a r t i c l e i n j e c t i o n . An integration s i m i l a r to that of the l a s t section then gives the t o t a l energy (Et): Er = cos(PI) _ [ E ~]cos(3Pl) [1-E Eref 1-E . m [sin(wmtf+Pm) - sin(Pm)] L w mt f . (2.8) For P1=Pm=wm=0, one gets 24 DEL= "1^ 1-E _J_.m=m/8 1-1 E (2.9) permitting, under these conditions, a modulation r a t i o 8 times as large as. for the modulation'of V1. This r e l a t i o n -ship holds, in fact, for a l l constant m curves drawn for P1=0. The constraints on the modulation of E may thus be taken from Figure 2.4 by multiplying i t s modulation r a t i o s by a factor of 8. 2.5 P3 Modulation Constraints In t h i s case the resonator voltage waveform becomes where the change i n the phase of the t h i r d harmonic with respect to the fundamental i s The t o t a l energy gain (Et) was obtained by integrating as before: Vr (t) = V1* (cos (P1) -E*cos (3*P1+P3 (t) ) ) (2.10) P3 (t) = Am*cos (wm*t + Pm) ( 2 . 1 1 ) E = cos (PI) - Ecos(3Pl) cos(A^cos5)dU + (2.12) E 0 w m t f Esin(3Pl) s i n C ^ c o s S )dy w m t f J*; 25 where = wm*t+Pm & = Pm = wm*tf+Pm The above integrals may be completed by using Bessel func-tion expansions to represent the integrands. The r e l a t i o n -ships used were: o o cos (u*cos (x)) = J0(u)+2*> (-1) n*J2n (u) *cos (2nx) (2.13) C O sin (u*cos (x)) = 2* ^{-1) n + l*J2n-1 (u) *cos ((2n-1) x) (2.14) n^l Using these expansions and then integrating gives the result = cos (PI). Ecos(3P1) (fr -\)Jn(Am) E 0 w m t f - 2Ecos(3Pl) y*'(-l)nJ2n(AmWsin(2nfr.) - sln(2nX-)) (2.15) w m t f 2n + 2Esin(3Pl) ( - l ) n + 1 J 2 r i - 1 (Am)• (sin((2n-l)Xf)-sin( (2h-l)"«t ) ) wmtf n t i 2n-l This expression may then be s i m p l i f i e d to ( 1 E ) - ^ = cos (PI) - Ecos(3Pl)»J0(Am) E r e f (2.16) + 2E f 1 ( - l ) n + 1 s i n [3P1 - (l-n)rtjTn (Am) (sin(nYf )-sin(nY>') ) w t f " L 3 n 26 from which the energy resolution may be calculated using Equation 2.18. To check the above equation, the l i m i t i n g case (P1=wm=0,Am<<1) was considered, giving the re s u l t (DEL/DELE) = ,952 * A m 2 (2.17) where DELR (=2/105) i s the desired energy resolution and Am i s i n degrees. For Am not small. Equation 2.16 was used to calculate DEL via the formula DEL =| E f " t r p f\ (2.18) Eref The r e s u l t s were plotted as a set of constant modulation amplitude curves. Figure 2.6 shows these plots for the beam phase (P1) equal to zero and the modulation phase (Pm) equal to 0° and 90°. Figure 2.7 shows the same curves but at a beam phase of -2°. From Figure 2.6 one can see that at P1=0° the magnitude of Am must be less than 1° for low f r e -quencies and less than 1.4° for frequencies of modulation higher than 4 kHz. From Figure 2.7 one can see that at a beam phase of -2° the r e s t r i c t i o n s on the modulation of P3 have become much more severe—now, f o r low frequencies of modulation. Am must be less than about .1°, while for f r e -quencies larger than 4 kHz. the magnitude must be kept under . -j * 27 a F i g . 2.6. P3 Modulated, P1=0° 2 . 6 P.1 Modulation Constraints I t was assumed that the fundamental phase ( P 1 ) of a p a r t i c l e varied as shown in Equation 2 . 1 9 . P(t) •= P0 + P 1 * C O S (wm*t+Pm) ( 2 . 1 9 ) Substituting the above in Equation 2 . 1 qives the time depen-dant resonator voltage waveform. The energy gain i s then obtained by performing the i n t e g r a l 2 8 0 . 0 E_fc = E„ 2 . 0 4 . 0 6 . 0 FREQUENCY (KHZ) Fig. 2.7. P3 Modulated, Pl=-2° 1_ cc w m t f J >; cos(P0 + Plcos&)dtf ft w m t f J j f i cos(P03 + P13cosy)d (2.20) where & =pm = wm*tf+Pm . P0 = phase of fundamental P1 = amplitude of modulation P03= 3*P0 P13= 3*P1 wm = freguency of modulation (rad./sec.) t f = t o t a l acceleration time. To perform the above integration the following integra-29 tion formula was derived: cos ( 9 +emcos"g)d}5 = (cose X(tfi - tff)Jo(0m) (2.21) - 2"g(-l) n + 1sin(0+(l-n)TC)J T 1(9 T T 1).(sin(n^ f) - sin(n8.)) n=l 2 n This formula was derived by f i r s t expanding the integrand of 2.21 and then substituting the i d e n t i t i e s 2.13 and 2.14 i n the res u l t i n g terms. The new integrand was then integrated term by term. The resultant sum of two i n f i n i t e series was then combined to give the new integration formula. Using t h i s formula to perform the integrations i n Equation 2.20 leads, after s i m p l i f i c a t i o n , to the res u l t (1-E) E t = Jo(Pl)»cos(P0) - E.Jo(3Pl)«cos(3P0) Eref - 2 y (-l)n+l( sin (nV f ) - sinCn^)) TsinCPO (2.22) wmtf fei n L + (l-n)^).J n(Pl) - Esin(3P0 + (l-n)«r> Jn(3Pl)~| 2 2 J from which the energy resolution was calculated usinq Equation 2.18. In the l i m i t as P0=wm=0,PK<1 i t was found that (DEL/DELR) = .00174*P1* (2.23) 30 where DKLH = 2/105 and PT i s i n degrees. This re s u l t was used to check. the computer evaluations of DEL/DELR. Figure 2 . 8 shows a set of constant amplitude (P1) curves f o r 2 - \ r — • 1 — 1 —I 1 P0=0. Since the beam p a r t i c l e s w i l l be injected i n t o the cyclotron at the proper phase, one need only consider the case where the modulation phase (Pm) i s equal to 90°. This Fiqure shows that f o r a beam phase (PO) of 0°, the amplitude of phase modulation (P1) must be kept under 6°. I t should be noted that, as the modulating frequency approaches zero, the r e s t r i c t i o n on the amplitude of modulation may be progressively relaxed. Figure 2.9 shows s i m i l a r curves at a beam phase of -2°. At thi s point the permissible magnitude of phase modulation has been reduced to 4.5° for frequencies larger than 3 kHz., while for frequencies between 1 kHz. and 3 kHz. the modulation should be kept under 4°. One can also 31 r n 0.0 2.0 4.0 6.0 8.0 10.0 F R E Q U E N C Y (KHZ) F i g . 2.9. P1 Modulated, Pm=90°, P0=-2° see that, as the modulating frequency approaches zero, the r e s t r i c t i o n s on P1 disappear. 2.7 Combination Of Errors To estimate the e f f e c t of simultaneous errors i n a l l four RF waveform parameters, consider the ef f e c t of small changes in these parameters w i l l have on Equation 2.1: Vr =MI&V1 + 2Y^E +2y_L&Pl + 2LYL&3 (2.24) 3V1 . OE BP1 3P3 Applying a summation of the form used i n Equation 2.2 then gives 32 A E T = 2q (jVr YAVICII) + ^ Vr VAE(II) + ^ Vr YAPI(II) + ^ VrYAP3(n)l LaVl N *>E N IVl^t 2)P3 N J (2.25) (2.26) Subsituting for the indicated derivatives leads to AE t = 2q[[cos(Pl) - Ecos (3Pl+P3)|^AVl(n) - Vlcos (3Pl+P3)^AE(n) - [ v i s i n ( P l ) - 3Esin(3Pl+P3^'V APl(n) + Vl-Esin(3Pl+P3)^]AP3(n) N . N For those ions crossing the accelerating gap at the peak of the fundmental SF (P1=0.0), the above Equation reduces to [l - Ecos(P3)]«£ AVl(n) - Vlcos (P3)^AE(n) N N (2.27) AE f c = 2q + 3Esin(P3)^APl(n) + Vl'Esin(P3)^AP3 (n) N N J For step changes in the parameters, the above summations can be replaced by N times the step changes; for time varying errors, each summation can be approximated by an integral of the type shown in Equation 2.3. Note that, because the wa-veforn i s f l a t ( "&Vr/dP1 = 0.0) for P1=P3=0°, Equation 2.27 cannot be used to estimate the useful RF accelerating phase-width. To do this, second order terms must be considered. 33 CHAPTER 3. METHODS FOR ACHIEVING WAVEFORM STABILITY In Chapter 2 the minimum tolerances on the parameters of the RF waveform were derived. These tolerances are the minimum required because an error i n only one parameter at a time was assumed; i n the actual system, a l l parameters w i l l have errors simultaneously. To reduce the errors i n the RF waveform, the following steps were taken: (1) removing, as far as possible, a l l sources of noise in the RF amplifier chains (2) reducing resonator tuning deviations to a minimum (3) using feedback to s t a b i l i z e the RF waveform. The reduction of the noise introduced into the RF chain has already been d i s c u s s e d 1 — t h i s i s merely a brief outline of the p r i n c i p a l causes. I t must be noted that, when highly stable operation i s required, the driver for the f i n a l power amplifiers i s operated in a saturated condition. By t h i s i t i s meant that the drive to the amplifier exceeds that value which produces the maximum output signal possible. At thi s point, small changes i n the driving s i g n a l amplitude produce no appreciable change i n the output. This leaves the f i n a l amplifiers as the pr i n c i p a l source of noise in the amplifier chain (control i s achieved by varying the screen voltages of i K. Brackhaus, Master's Thesis (Physics), pp. 20-31, University of B r i t i s h Columbia, 1972. 34 the I n i t i a l Power Amplifier) . A common source of noise i n amplifier c i r c u i t s , random thermal noise, i s not a problem at TRIUMF due to the high power l e v e l s used. Mechanical causes are, however, impor-tant. Thus, the detuning of resonant c i r c u i t s due to ther-mal expansion or mechanical vibrations can contribute s i g -n i f i c a n t l y to variations i n the RF output. The importance of t h i s effect varies with the Q of the c i r c u i t involved. This may be seen from Equations 3.1 and 3.2 which show the f r a c t i o n a l change in the output amplitude (jdA/A|) and the change in output phase (dP) of a simple resonant c i r c u i t when i t i s detuned, the amplitude of the input signal being kept constant. dw = W-WO w = driving frequency (rad/sec) WO = resonant frequency (rad/sec) B = bandwidth = HO/Q dA = change i n amplitude dP = change i n phase (rad.) Another mechanical cause of RF modulation i s the r e l a t i v e | d A/A | = 1/(1+2 (dS/B) 2 ) (3.1) dP = -2*Q* (dWO/WO) (3.2) 35 motion of an amplifying tube's elements (tube microphonics). i In this case the motion of the filament i s the principal source of problems. A f i n a l source of RF modulation in an amplifier chain is the variation of the level of. the plate and bias supply voltages, that i s , power supply ripple. This problem can be reduced by increasing the f i l t e r i n g in the DC power supplies. Using DC rather than AC power to heat the filaments will further reduce the modulation of the output. The TRIUMF RF resonators themselves contribute s i g n i f i -cantly to the variations in the accelerating voltage. The problem i s caused by the high RF Q of the cavity formed by the resonators: because this cavity i s excited at a constant frequency, small variations in, the resonant frequency of this cavity can produce large changes in the output (see Equations 3.1 and 3.2). The detuning is caused by thermal expansion and contraction (slow changes) and by the vibra-tion of the segment hot arms. To correct for the thermal effects, a pneumatic tuning system has been designed; to reduce the vibrations of the hot arms, two methods are used: i (1) reduction of the pressure fluctuations in the cool-ing water (the major cause of vibrations) by means of a pressure smoothing tank. (2) use of dynamic dampers to reduce the ab i l i t y of the hot arms to respond to fluctuations in the pressure of the cooling water. Measurements on the Center Region Cyclotron (CRC) have W n o D r t -M O 1-w rt-(C B AMPLITUDE LIMITER PHASE SHI PTE TUNING ERROR SIGNAL , U N_SELF^0SCILLATORY_MqpEJ *CT01 TJ? LPLEfO FREQUENCY SYNTHESIZER PHASE SHIFTER ROOT CURRENT|—* j_S AMPLE J PHASE DETECTOR OUTPUTS t t t RF ON/OFF PULSE MODE LOGIC DRIVE/OSC - * UNIT AMP. MOD. INPUT/SCREEN LT r - i JLOGIC ; •lUN]T_} r i .LOGIC J jJN IT _ , REFERENCE VOLTAGE •RESONATOR VOLTS INPUT INPUT FUNDAMENTAL SELECTOR MOD. RF AMPLIFIER LOGIC UNIT L I T j SCREEN MOD. FEEDBACK V I iAMPLIFIER LOGIC I UNIT ! R F SAMPLE F R E Q U E N C Y M E T E R TRANSMISSION LINE V O L T A G E | J I D E T E C T O R I RESONATORS PHASE DETECTOR T T R O O T LOOP T U N I N G P L U N G E R S PRESENT FUNDAMENTAL INPUT [ L O G I C ] L U N I T j TRIPLER P H A S E A N D A M P L I T U D E C O N T R O L 3 r d HARMONIC AMPLIFIER TRANSMISSION LINE RELATIVE AMPLITUDE AND PHASE METER RESONATOR WAVE-FORM SAMPLE ov 37 shown that the foregoing techniques are not, by themselves, sufficient to attain the degree of waveform stability de-sired. To attain the desired s t a b i l i t y , direct feedback control of the parameters of the BF waveform has been imple-mented. Five feedback systems have been designed1 (see Figure 3.1): (1) a system to regulate the amplitude of the fundamen-tal (23.075 MHz) BF signal. The error signal for this system i s derived.by comparing the .peak of the BF wave-form with a very stable reference. (2) a system to regulate the phase of the fundamental BF signal. The error signal is obtained by comparing the phase of the BF signal at the accelerating gap with the phase of the RF synthesizer. (3) a system to maintain the correct ratio (E) of third harmonic voltage. E i s measured and used to control the amplitude of the third harmonic RF signal. (4) a system to maintain the correct third harmonic-to-fundamental phase (P31). The phase of the third har-monic with respect to the fundamental i s measured and i s used to control the phase of the third harmonic BF signal. (5) a system to keep the resonators tuned to the funda-1 B.H.M. Gummer, BF Control—General Description, TBI-DN-71-29, August 1971. 38 mental RF frequency. The feedback sig n a l for t h i s system i s the deviation from a 90° phase difference between the current i n the coupling loop and the cur-rent in the shorted end of the resonators. The feedback control of the RF system w i l l be discussed i n d e t a i l i n Chapter 6. k p o t e n t i a l source of i n s t a b i l i t y — t h e interaction of the beam with the RF accelerating v o l t a g e — i s also discussed. 39 CHAPTER 4. RESONATOR VIBRATION DAMPERS The resonant RF cavity of the TRIDMF cyclotron i s com-posed of sections, one of which i s shown in Figure 1.4. While the "ground arm" portion of such a section i s fixed to the vacuum tank and i s therefore r i g i d , the "hot arm" por-tion i s a cantilever which i s free to vibrate. These vibra-tions detune the high Q (Q = 6000) RF cavity, and consequen-t l y t heir amplitude must be - kept to a minimum i f the RF power required to obtain the desired accelerating voltage i s not to exceed the c a p a b i l i t i e s of the RF amplifiers. An insulator could not be found which was suitable for connec-tin g the hot and cold arms; hence, dynamic damping i s the only available means for d i r e c t l y reducing the vibrations of the hot arm. The primary cause of these vibrations i s the flow of cooling water through the channels i n the hot arm skin. 40 4.1 Damper Model 4.1.1 Description Of Model The theory of a dynamic damper i s known1,2 and w i l l only be summarized here. At th e i r fundamental resonance, the hot arms may be modelled as shown i n Figure 4.1 by appropriately choosing the parameters M1, K1 and C1. These quantities may be expressed i n terms of the measured spring constant, res-onant frequency and Q of the panel vibrations. For the hot arms: K1 =measured spring constant (lbs./in.) at point where damper i s to be attached = 30.303 l b . / i n . M1 ^ e f f e c t i v e mass of the panel =K1/S12 =0.033595 lbm. «1 ^measured resonant frequency =4.78 Hz =30.0336 rad./sec. C1 =coef. of viscous f r i c t i o n =(M1*K1) Q1~:1 = .003386 lb . s e c . i n - * Q1 =measured Q of the panel vibrations = 300 For the system in Figure 4.1, the Laplace transform of the r e l a t i o n s h i p between the dr i v i n g force and the displace-ment of the main mass (hot arm) may be written as 1 S. Timoshenko, Vibration Problems i n Enjjijieerincj,. Van Nostrand Inc., 1955, p.210. 2 J. Ormondroyd and J. P. Den Hartog, The Theory of the Dynamic Vibration Absorber, Trans. A.S.M.E. Vol. 50, p. APM-241, 1928. C l > Main Systen c 2 Damper Fig. 4.1. Main System Plus A. Dynamic Damper F(s) = Ml * s 2 + Cl*s + K l + 1 + 1 C2+B2*s M2*s2 x(s) (4.1) which then gives the transfer function M2*s2 + C2*s + K2 p Z <S V ~ (Ml*s^+Cl*s+Ki) (M2*s^ +C2*s+K2)+M2*s^ (K2+C2*s) {*' 42 4.1.2 Optimal Choice Of Damper Parameters Since the derivation of the optimal damper parameters i s given i n l i t e r a t u r e , t h i s w i l l be merely a summary of the relevant equations. The following variables are f i r s t de-fined: Xst = F/K1 = s t a t i c d e f l e c t i o n of the hot arm by a force (F) H2 = K2/M2 = natural frequency of the absorber (rad./sec.) P = H 2 / M 1 =ratio of the masses i n the system S = W 2 / W 1 =ratio of the two natural frequencies "(5 = W/§n =ratio of the frequency of the applied force Jci-c-e1. to the natural frequency of the main system. yU. = C 2 / ( 2 w 1 * J 3 2 ) defines the damping. Using the above parameters, the magnitude of the res-ponse squared may be written as: 4 / V$ -1 + pIS j + [fi V - (5 - •)(* - I )] This equation i s derived from Equation 4.2 by assuming negli g i b l e damping (C1=0.0) i n the main system. The design method may be summarized as follows: (1) f i n d fl1 and W1 (2) choose the largest p r a c t i c a l r a t i o P =M2/M1 (3) use $ = 1/{1 + p) to find the natural frequency the 43 damper must have (W2 = Wl) (4) f i n d the spring constant (K2) of The above data i s for the fundamental 'hot arm mode, using viscous f r i c t i o n . Table 4.1. Optimum Hot Arm Damper Parameters absorber: K2=H2 * ( S w i ) 2 (5) f i n d the required damping (C2) : C2=2*W 1 *M2*y^ where: the M2 (lbm) K2 (lb/in) C2 (Ib.sec/in) f2 (Hz) A i .50 1.68xl0~2 6.73 .238 3.19 .20 6.72xl0 - 3 4.21 8.41xl0 - 2 3.98 .10 3.36xl0 - 3 2.50 3.39xl0~2 4.35 .05 1.68xl0~3 1.37 1.28xl0~2 4.55 .02 6.72xl0 - 4 .58 3.39xl0 - 3 4.69 > =3 ft y 8 (1 + p)5 (4.4) Once H1 and W1 have been defined, one parameter , ^ , must be chosen a r b i t r a r i l y . The reason for t h i s i s that there i s no optimum value for ^ since the amount of damping i s a mono-t o n i c a l l y increasing function of t h i s v a r i a b l e . I t s choice must therefore derive from other considerations. 44 The optimum damper u t i l i z e s viscous f r i c t i o n ( f r i c t i o n force= constant*velocity), but t h i s type of f r i c t i o n i s not always p r a c t i c a l to implement. Because th i s i s the case for the hot arm dampers, s l i d i n g f r i c t i o n ( f r i c t i o n force = constant*sign(velocity)) i s u t i l i z e d instead. For a given amplitude (A) and freguency of vibration (w), a damper u t i -l i z i n g viscous damping may be replaced by another type i f i t dissipates the same amount of energy per cycle. Thus, i t i s found that for s l i d i n g f r i c t i o n Ff = . 25*C2*A*w*T£ (4.5) where Ff i s the required f r i c t i o n a l force. Note that a damper employing t h i s type of f r i c t i o n i s optimal only for a s p e c i f i c amplitude and frequency of v i b r a t i o n . Ff for the hot arm dampers was found by adjusting a prototype damper u n t i l the vibrations of the hot arm were minimized. Table 4.1 gives the calculated optimum damper parameters for v a r i -ous mass r a t i o s . 4.1.3 Predicted Reduction In The Amplitude Of Vibration \ 45 4.1.3.1 Excitation At A Single Fxeguency I f the system which i s to have the amplitude of i t s vibrations reduced i s excited at a single frequency, then evaluating the magnitude of i t s response with and without a damper w i l l give the reduction in the amplitude. Of p a r t i -cular i n t e r e s t , of course, i s the reduction of the peak i n the response of the undamped system. If an optimal damper (defined in Section 4.1.2 ) i s used, t h i s reduction may be written as: Xs/Xd = Fro* (M1+K1) 2 / C l = Q1* (p/I 2+p])2 (4.6) Here Xs i s the response of the main system at resonance and Xd i s the response (see Figure 4.2) at the f i r s t peak that r e s u l t s when an optimal damper i s added while keeping the magnitude of the ex c i t i n g force the same. Q1 i s the me-chanical Q of the main system and FrO i s a factor which de-pends only on the mass r a t i o (Rm) : -h Fro = £ 1 .+ 2*Rm ] (4.7) (Rm = M1/M2) Figure 4.3 gives a plot of FrO versus Rm. For our system, using a mass r a t i o of 20 to 1, one finds that the peak of the undamped response should be reduced by a factor of 46. 41 • C o o. tn o 1 60 DAMPER SYSTEM PARAMETERS ...... p l 0 T „ n > !».«,,, x i • ( i . i i s '>5r - o i C I * 0 , 3 3 t l W 0 i K l • 3 0 , 3 0 3 M i • « , i k 7 <)7r '0 i .5 C o o p t i m u m « i • l ' . » 7 « i . . . . . . C l O T N O . ? . . . . . . « i • o . u s o s r - o t C l • 0 , 3 3 f K > A E * 0 i K l • 3 0 . 3 0 3 " i • 0 , t l i 7 < ) 7 F - 0 ' c i « o . i j u u i r - o i " i « 1 . 3 7 0 3 . C2 o p t i m u m . . . . . . P I O T MO. ] » • » • • • M I • t . u s q s r - o t C l • 0 Kf-tl 2 K l • 3 0 . 3 0 1 M i • o . i k 7 9 7 E - 0 i 2 . 0 C , o p t i m u m c i • o,?S6aiE-oi « K i • 1 . 3 7 0 3 PLOT SYMBOLS!! Til I0> ARC I 0.•,»,•,'..A,0,A,0,« I » - « . « . « O « * V t * . M . M . » . M / * M <0,« K /« M m M •» M K- M . t » . M . t J M i M . M i « * . « » Hk M i Mn M . t M B > MM M . M i Mi M . M k —. M t « , a. . \//r m J it I 9 J 1 \ \\ \ \ \ \ I * O X « o m * © Frequency (liz.) o «tr • •» i O N i/lC M ) i/i <r AI « v • tr o- oo o • i o M « m KtN^tOI/lffttffi — a ^ r ^ - C L n i ^ c r - o - r v I T KT I P » « KT cr ar < kft •« r C - <D Kl IM • U> M » MB #n c •ODI/IIW C O O 0.0 0.5 1.0 1.5 2.0 L0G1CHM1/M2) F i g . U.3. Plot Of FrO Versus Log 10 (fim) 4.1.3.2 White Noise Excitation When the vibration of the hot arms i s excited by a sto-chastic s i g n a l ("band limited White Noise" as f a r as t h i s system i s concerned), then the previous c a l c u l a t i o n of the amplitude reduction does not apply. That i s , one must con-sider the entire spectrum of the input, not just one f r e -guency. Knowing the spectrum density of the dri v i n g force (Si ( f ) ) , one may calculate the spectrum density of the re-s u l t i n g vibration (Sv (f)) and consequently the RMS value of the hot arm d e f l e c t i o n . The output power density spectrum i s given by 48 Sv (f) = | Z (f) | 2 * S i (f) (4.8) where Z(f) i s the transfer function defined i n Equation 4.2. The RMS value of the response i s therefore Xrms = { |Z (f) | 2 * S i (f) *dfp (4.9) One may assume a constant input spectrum density (Si (f)) i n the frequency region where a s i g n i f i c a n t hot arm response i s possible. Hence, a c a l c u l a t i o n of the reduction in the RMS response involves a c a l c u l a t i o n of the area under the square of the response curves shown i n Figure 4.4. The area under the curve of the square of the undamped response has been c a l c u l a t e d : 1 Ao = .25*[C1*K1] (4.10) However, an exact analytic expression for the i n t e g r a l of the damped response squared has not been found. Instead, the i n t e g r a l was evaluated numerically for the TRIUMF dam-pers and approximated for the general case. I f the i n t e g r a l i s approximated as shown in Figure 4.4, then the reduction in the RMS amplitude (Rv) may be written as JC. K i t t e l , Elementary S t a t i s t i c a l Physics. Wiley, 1958, p.147. 4 9 -0.25 -0.125 LOG10(F/F0J 0.125 Fig. 4.4. Amplitude Of Vibration Squared: Hot Arms poo ' . R v 2 = {4*C1*K1*J | Z (f) | 2*df} " 2 (4-11) where |Z(f)| 2 i s approximated by |Z(f ) | 2 = Ae (1+Be*f 2 ) - i + A {1+ B (f-f 1) 2) - 1 + A(1+B(f-f 2 ) 2 ) - i . (^.12) 50 0 . 5 1 . 0 L O G 1 0 ( M 1 / M 2 J F i g . 4.5. Sv: Approximate And Exact, Fund. Mode The six parameters of the l a s t equation are defined as f o l -lows: a) f 1 = ( b) f 2 = ( r 1 1 . 1 +fi 1 Li 2+(3_ 51 c) Ae = K1 -2 d) Be = f 1 ~ 2 e) A - .5{Xc + [Xc 2- 3*Xd*Xb/2] J2 } f) B = {8A/Xb - 4} {f2 - f 1 } - 2 where fo i s the resonant frequency of the main system and Xd i s the magnitude of the peaks of | Z ( f ) | 2 £= {1 + 2/|S j le}. Xb i s the magnitude of J Z ( f ) | 2 at a frequency half-way be-tween f1 and f2 Xb = jZ[ (f2+f 1)/2]! 2 (4.13) and Xc i s equal to Xd + Xb/4. The int e g r a l of Equation 4.12 was found to be Z ( f ) 2 d f = - 7--I5 + + tan-Kf 1 B V tan-Hf2Bl>3 (4.14) 0 2 (B e) 2 ( B ) 2 (B)' For values of p smaller than 1/10, the l a s t eguation was found to give values of Rv that d i f f e r e d by less than 2% from the resu l t obtained by a computer integration of 52 poo | Z ( f ) | 2 d f . If a larger error can be tolerated, then o ' Aeo/(Y ao<: • A Equation 4.13 may be approximated by R — + c- . 2 (B e) 2 ( B ) 2 Using t h i s equation, an error of 8% was observed i n Ev f o r (3=1/10. This error was found to decrease with decreasing p , becoming about 5% for ^ =1/50. Figure 4.5 compares the reduction predicted using the approximations and the reduc-tion predicted using a computer integration of | Z ( f ) | 2 . Note that, using a mass r a t i o of 20 to 1, the l a t t e r calcu-l a t i o n predicts that an optimal viscous damper would reduce the amplitude of the hot arm vibrations by a factor of about 5.8. This reduction i n amplitude i s not a rapidly varying function of the mass r a t i o . I t i s also useful to know how s e n s i t i v e the e f f e c t of a damper i s to changes in i t s parameters. This was found by varying M2, K2 and C2 i n Z (f) and then finding the new value of the i n t e g r a l : | Z ( f ) j 2 d f . Figure 4.6 displays the re-0 s u i t s for the damper designed to cope with the fundamental v i b r a t i o n a l mode of the hot arms. It i s i n t e r e s t i n g to note that the minimum RMS response occurs f o r a value of C2 that i s about 20% lower than that required f o r maximum peak redu-c t i o n . Also, i t i s s i g n i f i c a n t that the response i s less s e n s i t i v e to variations i n C2 than to variations in M2 and K2, for t h i s i s the parameter that i s most d i f f i c u l t to re-produce accurately. 53 4.2 Measurements 5 4 4 . 2 . 1 W i t h o u t D a m p e r s A s e r i e s o f m e a s u r e m e n t s w e r e c a r r i e d o u t t o d e t e r m i n e t h e v i b r a t i o n a l c h a r a c t e r i s t i c s o f t h e r e s o n a t o r h o t a r m s . 1 I Vibrator \ \ \ N\x 4- A n n H . f i or \$ Hot Arm vibration pick-up |.(y) Vibratiop-A'mplitud V i b r a t i o n fc Meter v i b r a t i o n frequency F i g . 4 . 7 . F r e q u e n c y R e s p o n s e M e a s u r e m e n t Two b a s i c a l l y d i f f e r e n t t y p e s o f m e a s u r e m e n t w e r e m a d e : (1) a f r e q u e n c y r e s p o n s e m e a s u r e m e n t — i n t h i s c a s e t h e f r e q u e n c y o f t h e e x c i t i n g f o r c e w a s v a r i e d a n d t h e r e s -p o n s e w a s p l o t t e d ( s e e F i g u r e 4 . 7 ) . (2) a f r e q u e n c y s p e c t r u m m e a s u r e m e n t — i n t h i s c a s e t h e e x c i t a t i o n w a s d u e t o t h e f l o w o f c o o l i n g w a t e r t h r o u g h t h e c h a n n e l s i n t h e h o t a r m s k i n . T h e r e s p o n s e s p e c -t r u m w a s m e a s u r e d a n d p l o t t e d u s i n g a B r u e l a n d K j o e r i S e e a l s o : M. Z a c h S E . W. B l a c k m o r e , P r o t o t y p e R e s o n a t o r T e s t s , T R I U M F F i l e 6 2 . 4 , J u l y 3 , 1 9 7 0 . 55 C o o l i n p w a t r r flow Strip Recorder Hot Arm Vii:rati on pick-up "O Spectrum Analvser V i b r a t i o n Meter ( l i n e a r response) 4-F i g . 4.8. Frequency Spectrum Measurement spectrum analyser (see Figure 4.8). 2 The frequency response measurements ' revealed that a single hot arm has the two basic modes of vibration shown i n Figure 4.9. Unfortunately, t h i s plot cannot be used to c a l -culate the Q of these modes: to get f u l l development of the peaks would have required a p r o h i b i t i v e l y slow frequency sweep rate. Separate measurements (decay time constant) gave a Q of about 300 for the fundamental mode and a some-what lower Q (=250.) for the torsion mode. This mode struc-ture i s altered when the hot arms are connected at t h e i r t i p s ; Figure 4.10 shows the mode structure r e s u l t i n g when four hot arms are connected at the i r t i p s with the e x c i t a -tion applied to the middle of the second hot arm from one 2Since the lowest resonance of the hot arms (4.7 Hz.) was below the lowest frequency range of the spectrum analyser (20 Hz.), a balanced modulator (suppressed carrier) was b u i l t to translate the response spectrum to a range mea-surable by the analyser. A 20 Hz. c a r r i e r frequency was used. 58 end. The two modes shown i n the frequency response measure-ments also appear in the spectrum plots. Figure 4.11 shows a t y p i c a l response spectrum for one hot arm when the water flow rate i s 12 gpm (city water supply). Figure 4.12 shows the response when 3 hot arm panels are loosely connected at t h e i r t i p s . An estimate of the effect of reducing the water flow rate on the v i b r a t i o n a l l e v e l of the hot arms may be made by considering Figure 4.13 which shows the spectrum which resulted when the water flow rate was decreased to 5 gpm. (Note that an increased paper speed was used for t h i s plot.) 4.2.2 With Dampers A damper with a 20 to 1 mass r a t i o was designed using the method outlined in Section 4.1.2.1 Figure 4.14 shows a drawing of the f i n a l design. Note that a .015 inch thick phosphor bronze l e a f spring i s employed to give the required spring constant while retaining l a t e r a l r i g i d i t y . The mass of the damper i s provided by a lead f i l l e d section of 1"x2M aluminum channel stock, while the f r i c t i o n i s provided by a spring mounted ceramic button which rubs on an adjustable s t a i n l e s s s t e e l plate. Since t h i s i s s l i d i n g f r i c t i o n , the damping provided by t h i s button varies with the amplitude of vibration. Consequently, the f r i c t i o n must be adjusted f o r *See TRIUMF drawings D-963 and D-964. Bruel & Kjaer Copenhagen Frequency (Hz.) 1. 2. 3. 4. Conditions of Measurement 16 gal/min coolant flow vibration pick-up mounted on front corner of panel no dampers 20 Hz. carrier used to translate frequency of signals into range suitable for the spectrum analyser Bruel & Kjoer spectrum analyser settings: (a) 6% bandwidth (b) 2 mm/sec writing speed (c) .3 mm/sec paper speed (d) RMS response Fig. 4.11. Vibrational Spectrum: Single Hot Arm Bruel & Kjoer Copenhagen Fig. A.12. Vibrational Spectrum: 3 Hot Arms, 16 GPM 25 db 90 Bruel & Kjcer Copenhagen --J ,A ft 1 Hi 1 A f\ f, Su H J 1 H K *• ,i ,." i A i 1 1 1 *H 1 »t ft - . i x i • i i I ' p. I I ': 1 \ V -IS L O ' •A 10 10 Frequency (Hz.) 20 30 40 Same c o n d i t i o n s as f o r F i g . 4.11., except that the water (coolant) flow r a t e was reduced to 5 gal/min and spectrum analyser paper speed was increased to 1 mm/sec. F i g . 4.13. V i b r a t i o n a l Spectrum: 3 Hot Arms, 5 GPU Fig. 4.14. Drawing Of Damper For Fundamental Mode the amplitude of vibration which w i l l be experienced by the hot arms under operating conditions. The two prototypes that were constructed—one for the fundamental mode and one for the torsion mode—were therefore adjusted for maximum damping with a coolant flow of 16 gpm. The same adjustment was then used for the remaining dampers. Figure 4.15 shows the spectrum response when a 4.7 Hz. damper i s employed. Figure 4.16 shows the re s u l t s when both a 4.7 Hz. and a 10 Hz. damper are employed. I t i s im-mediately apparent that, as shown in Section 4.1.3.2, the reduction of the resonance peaks i n the spectrum response i s much less than that for single frequency . excitation. In f a c t , i t was found that the use of these dampers reduces the RMS amplitude of vibration by a factor of only about 2. Figure 4.17 shows a sample of the hot arm vibrations with and without these dampers. It should be noted that the orientation of the long axis of the dampers with respect to the long axis of the hot arms was not found to a f f e c t damper performance. Bruel & Kjoer Copenhagei •a db 20 20 c/s 25 30 40 50 0 10 20 30 40 Frequency (Hz.) • Same c o n d i t i o n s as f o r F i g . 4.11., except that the paper speed of the spectrum analyser was 1 mm/sec. i F i g . 4.15. V i b r a t i o n a l Spectrum: 4.7 Hz. Damper Briiel & Kjcer 2C Copenhagen <l tt 1 ye •u 1 1, -~ I v_ i Ii . i .1 ' ' ' iii 1 i. 1 r ft. 1 v f i ! t. h ff 1 Pi i i 1 L. i P \ t • I h 1 v-5 \ I I P3 20 c/s 25 30 40 5 10 20 — Frequency (Hz.) • 50 60 63 30 AO Same c o n d i t i o n s as f o r F i g . 4.11. 4.16. V i b r a t i o n a l Spectrum: 4.7 Hz plus 10 Hz Dampers F i g . 4 . 1 7 . Hot Arm Vibration With/Without Dampers 67 CHAPTER 5 . PNEUMATIC TUNING SYSTEM The system to be kept tuned i s the resonant, c o a x i a l , 1/4 wave cavity used in the TRIUMF cyclotron. I t i s important that t h i s cavity i s kept tuned to the RF driv i n g frequency to minimize the power required to achieve the required res-onator voltage. Loop coupling i s employed to feed power into t h i s res-onant RF structure — when the resonator i s tuned to the d r i v -ing frequency, the current i n the loop i s 90 degrees out of phase with the current i n the root (shorted end) of the res-11EF = 0.0 i'neunatic Line Driverf Feedback Amplifier Pneumatic Lines W Bellows [Tuning Error] Detector + AF F i g . 5 . 1 . Block Diagram Of The Resonator Tuning System onator. Thus, to provide an error s i g n a l for the TRIUMF resonator tuning system, the phase of the current i n the loop i s compared with the phase of the current i n the root. The deviation from a 90 degree phase difference i s propor-68 t i o n a l to the difference between the d r i v i n g frequency and the resonator resonant frequency (for small deviations). Figure 5.1 i l l u s t r a t e s , in block form, the organization of the tuning system. Figure 5.2 shows the actual tuning me-chanism: the volume of the resonators i s varied by means of pneumatically driven tuning f o i l s . There i s one pair of these tuning f o i l s per resonator rootpiece and each pair i s actuated by a pair of opposing bellows. These bellows are organized into 16 i d e n t i c a l groups of f i v e pairs each. Eight p o r t s — 4 top,4 bottom—through the cyclotron vacuum tank provide access for the a i r l i n e s which drive these bel-lows. 5.1 System Transfer Functions 5.1.1 Bellows Transfer Function 5.1.1.1 Pressure-Displacement Transfer Function In order to calculate the change i n the , resonant f r e -quency of the resonators, the displacement of the tuning bellows must be known. To accomplish t h i s the bellows were modelled as shown in Figure 5.3. Equating the forces on the model gives: M*X = Kf*X + Ks*X + Pb*Ab (5.1) 69 I RESONATOR ROOT PIECE MOVEMENT 1^ \LvvwJI BELLOWS r AIR LINES SUPPLYING BElLOWS .538" = La TUNING FOILS 32' MOVEMENT •TUNING FOIL Lb = 1.125" 4.050' SECTION A-A' ( TUNING FOIL ) F i g . 5.2. Schematic Of Pneumatically Actuated Tuning F o i l s 70 where: M ^ e f f e c t i v e mass of the tuning mechanism (lbtu.) X =displacement of the bellows from t h e i r equilibrium position (in.) K f ^ c o e f f i c i e n t of viscous f r i c t i o n (represents the d i s -sipation in the system) (lb./in.) Ks =spring constant of the system (lb./in.) Pb =deviation of the bellows pressure from the mean value (psi.) Ab =effective area of bellows (in 2) Laplace transforming Equation 5.1 then gives the required transfer function: X(s)/Pb(s). = (Ab/M)[s 2 + (Kf/M)s + (Ks/M) ] " i (5.2) Fi g . 5.3. Tuning Bellows Model 71 = AbO[Wb-2*s2 + (2*Eb/Wb)s + 1]~ l (5.3) where: AbO = Ab/Ks = low frequency response (in./psi.) Wb =\JKs/M = resonant frequency (rad./sec.) Eb =. 5Kf \ / K S * M ' = damping factor Measurements indicated that Eb=.55 and that Wb=12 Hz. Another set of measurements on a tuning bellows using thinner (.003") f o i l s was subsequently made. The r e s u l t s of these measurements are plotted i n Figure 5.4; according to t h i s graph we have Eb=.5 and Hb=5 Hz. Eb and Wb are s u f f i c i e n t to characterise the transient response of the bellows; however, to fi n d the actual magni-tude of the response, one must know AbO. I f the pressure i n both bellows of a pair i s simultaneously and oppositely varied, then Ab0=,24 i n . / p s i . ; i f the pressure i n only one bellows i s varied while the other opposing bellows i s sealed, then AbO i s less than, or equal to .12 i n . / p s i . 5.1.1.2 Pneumatic Input Impedance Of The Bellows In order to calculate the frequency response of the tuning system, the pneumatic input impedance of the bellows must be known. This i s derived by f i r s t assuming an i d e a l gas equation (P*V=Mg*R*T) and then d i f f e r e n t i a t i n g with res-pect to time: 6 7 8 9 ' FREQUENCY (HZ) 73 l d P •+ l d V - l d T = i_dMg_ (5.4) P dt V dt T dt Mg dt where P = pressure (psi.) V =volume (in 3) Mg = mass of gas (Ibm.) i n bellows. 1-n To f i n d dT/dt a polytropic gas equation (T*P n = const) i s assumed, which when d i f f e r e n t i a t e d gives T - i * (dT/dt) = (n- 1) *n-**P - i * (dP/dt) (5.5) (n=Cp/Cv=polytropic exponent) Since small pressure fluctuations were assumed, the gas den-s i t y may be taken to be constant--this leads to the resu l t that dMg/dt = Ib*Mg/V (5.6) where lb i s the bellows input flow rate ( i n 3 / s e c ) . Hence, since dVb/dt = &b*[dx/dt], the bellows response i s V*n- 1*P~ 1* (dP/dt) + Ab*(dx/dt) = l b (5.7) For small signals, take V and P in the c o e f f i c i e n t as being approximately constant, that i s , 74 V*n - i*p - i = Vo*n - i*Po~ i (5.8) where Vo and Po are mean values. , Hence, the (Laplace trans-formed) equation becomes: Vo*n-i*Po-**s*P (s) + Ab*s*X(s) = lb (s) (5.9) Since the pneumatic input impedance of the bellows i s defined as Zb (s) =Pb (s)/lb (s) one f i n d s , using Equations 5.2 and 5.9 that: FKfl Ks Zb(s) = Kb (S2 + LM . s + M ) Ab2*s (s2 + ["Kf" s + Ks+Kb ) LM . M (5 . 1 0 ) 2«E1 s Kb*Ks 1 ( Lll + wl ° + ' 1 ) (5 . 1 1 ) Abz(Ks+Kb) s ( j"s ^  + 2<E2JL& + TT w2 where: Kb = pneumatic sprinq constant of the bellows = n*Po*Ab2/Vo (lb./in.) H12 = Ks/M (rad./sec.) 2 W22 =(Ks + Kb) /M (rad./sec.)2 E1 = Kf/2* \/(Ks*M) E2 = Kf/2* ( (Kb+Ks) *M) Figures 5.5 and 5.6 show the magnitude and phase of Zb(jw) for the parameters used i n t h i s analysis. From Equation 5.11 i t can be seen that f o r both low and high f r e -quencies the input impedance becomes capacitive. For low frequencies the bellows may be approximated by a spring loaded volume with impedance Zb (low f req) = Kb*[ jw*Ab2* (1 + Kb/Ks) ] — 1 (5.12) high frequencies the bellows do not respond mechanically may therefore be approximated by a fixed volume: Zb(high freg) = n*Po*[ jw*Vb (5.13) Note that for an intermediate frequency the bellows impe-dance can become inductive: t h i s e f f e c t (shown i n Figures 5.7 and 5.8) may be used i f an inductive pneumatic impedance i s required to "tune" the pneumatic l i n e s . To achieve an inductive impedance at some frequency (w)., one must choose device parameters so that w K < w « w 2 . 5.1.2 Transfer Function Of A Pneumatic Transmission Line The transfer matrix used in this analysis i s based on equations derived by C. P. Rohmann and E. C. Grogan (Trans. A.S.H.E., May 1957). Small signals are assumed, that i s , i t "17,11 - 8 , 6 7 C O 0 , 6 7 1 7 , 4 4 1 """ X I X I X MAGNITUDE ( d 9 ) e ~ X X I X I X 1 X I X I X I X X X I X J X 1 X I X 1 X H- 1 ( x TO I X J X U l I X I x U l I x I x • • I x M 2 • I X 3 d I x •O 05 C 3 X X r t H* I x rt I x f c x x 3 O . •a re I x <B I x c o I X (1) H> I X 3 o c= X X n n> I X I X M I X o IX X XI X I X I X I X I X I X I X I X I X I X I X I X I X I X I X . 1 X I X I X I X I • X x It I ' x • X X I X I X I X 1 X ' I X I X X X " X X I X I X X X I X X X X X ; ' 'X X X X X , X • X ' X X X X X I X I X X X I X I X I X 1 X '. x 1 { • I t 3 0 m o 1= rn H O - < 0 , 10 0 0 0 , I i>50 0 , 1 1 0 ? 0 , 1 15S 0 . ! ? 1 6 0 , i nt. 0 , 1 ; «l 0 0 , 1 1 . 7 0 , 1 1 7 7 0 1 5 5 1 0 | o 2 9 0 1 7 1 0 0 1 7 1 6 0 IPtb o I V r O 0 2 0 7 9 0 2 1 6 3 0 2 2 1 2 0 2 * 0 7 0 2 4 2 7 0 2 6 5 3 0 2 7 8 6 0 2 9 2 5 0 3 0 7 2 0 3 2 2 5 0 3 3 8 6 0 3 5 5 6 0 3 7 3 3 0 3 9 2 0 0 1 1 1 6 0 1 3 2 2 0 0538 0 0765 0 SOC? 0 5 2 5 3 0 5 5 1 6 0 5 7 9 2 0 6 0 3 1 0 6 3 S 5 0 6 7 0 5 c 7 0 4 0 0 7 3 9 ? 0 7 7 6 2 0 6 1 * 0 0 e s i 7 0 6 9 3 5 0 9 4 3 0 0 9 9 0 6 1 0 0 0 1 1 0 9 2 1 1 1 0 * 7 1 2 0 o l 1 2 6 0 5 1 3 2 7 S 1 3 9 3 9 1 0 6 3 6 1 5 3 » 7 1 6 1 3 6 1 69uJ> 1 7 7 9 0 1 6 6 7 9 t 9 6 1 3 2 0 5 9 0 2 1 6 2 3 2 2 7 0 0 2 3 6 0 0 2 S 0 3 2 I 6 2 6 3 2 7S97 2 8 9 7 7 3 0 0 2 6 3 1 9 « 7 3 3505 3 5 2 2 2 : 6 9 S 3 } 6 S 3 2 « 0 7 7 0 2 M S a 0953 7 2 0 1 « 9 5 6 1 s 2 0 3 9 5 0 6 0 ) 5 7 3 7 3 6 0 2 . 1 t 3 2 5 0 6 6 0 1 6 6 9 7 37 7 S 2 2 0 7 6 * 6 5 ft 07/19 6 0 7 6 6 K 9 0 0 0 9 >'50 9L 77 6 D006 9 9 6 ? 1 0 9 1 i>?2( 1 / w», 9 91l>9 9 t>52f 9 !l>20 9 U l 1 5 1 l>9» S i f 0 ? 5 1956 ft 102/. & i56r> 0 17 HZ/.0 ft 2f 99 £ 1969 I 2?21 t SnSt t t t J 6 t E 92H0 f 1/.69 2 169/ 2 £929 2 2105 ? ougr 2 001? 2 t ? 9 i 2 0650 2 ( 1 9 6 1 619H I 0 6 / . 1 1 2t!69 I 91 19 1 i « $ S t 9£9I> I 6f 6 f I snt I £,"92 I | p g ? I i 9 r i I 1263 I l o r o t 9066 0 Of »6 0 5969 0 tS50 0 0519 0 29/1 0 2 6 £ i 0 OtiOl 0 5019 0 SdJ9 0 1909 0 2615 0 9IS5 0 JS25 0 1005 ' 0 59i» 0 9f Sr> 0 2 2 £ » ' 0 911" ' 0 0?6f " 0 i f i f ' 0 955f ' 0 9«Jf •o s?2r •o 2 i o r ' 0 S26? •o 9912 ' 0 SS9? •o 125? ' 0 /.Oft? •o 262? ' 0 £J1? ' 0 6/0? 0 C96I ' 0 9991 •o 9 h l t ' 0 0 WI •o 6291 ' 0 IC5! •o U M •o IOM •o 0»l 1 ' 0 '121 ' 0 9 t ? | •o 4 <'. 1 t "0 20 11 'o C50! ' 0 ooct ' 0 N X > o z L U 3 o lli CC 1 6 * 9 9 -F R E Q U E N C Y ( H Z ) F i g . 5.7. Magnitude of Bellows Input Impedance (Inductive) rt •90.00 0,0 I I S , 0 0 90,00 JO m o c m z o -< X N o . i o o o 0. I O S 0 0 , 1 1 0 ? 0,115s o.ieit 0. 1 2 7 6 o , l S o o 0 . 1 9 0 7 0 . 1 9 / 7 0 , 1 5 5 1 0 , 1 6 2 9 0 , 1 7 1 0 0, 1 7 9 ( , _ 0 , l « S 6 o , l 9 « o 0 , 2 0 / 9 0 . 2 1 6 1 0 . 2 2 9 ? 0 . 2 1 0 7 0 . 2 S 2 7 0 . 2 6 5 3 0 , 2 7 6 6 0 , 2 9 2 3 0 , 3 0 7 2 0 , 3 2 2 5 0 , 3 3 6 6 0 , 3 5 5 6 0 , 3 7 3 3 0 . 3 9 2 0 0 , 9 1 1 6 0 . ' 3 2 2 0 , 9 5 3 6 0,9765 0 , 5 0 0 3 0 , 5 2 5 3 0 , 5 5 1 6 0 . 5 7 9 2 0 , 6 0 6 1 0 , 6 3 6 5 0 , t 7 0 5 0 . 7 0 9 0 0 , 7 3 9 2 0 , 7 7 6 2 O . d l S O 0 . 6 5 5 7 0 , 6 9 6 5 0 , 9 9 3 9 0 , 9 9 0 6 1 , 0 9 O | I , 0 9 2 1 1 . 1 9 6 7 1 , 2 0 9 1 1 , 2 6 0 3 1 , 3 2 7 5 1 , 3 9 3 9 1 , 9 6 3 6 1 . 5 3 6 7 1 , 6 1 3 6 1 . 6 9 9 2 1 . 7 7 9 0 1 , 6 6 7 9 1 , 9 6 1 3 2 , 0 5 9 0 2 . 1 6 2 3 2 , 2 7 0 0 2 , 3 6 0 0 2 , 5 0 3 2 2 , 6 2 6 3 2 , 7 5 9 7 2 , 6 9 7 7 3 , 0 0 2 6 3. 1 9 0 7 3,3505 3 . 5 2 2 2 3 . 6 9 6 3 3 . 6 6 3 2 9 , 0 7 7 0 0 , 2 6 1 3 0 . 0 9 5 J 1 . 7 2 0 1 9 , 9 5 6 1 5 . 2 0 3 9 5 . 0 6 0 1 5 , 7 3 7 1 6 . 0 2 9 1 6 , 3 2 5 0 6 , 6 0 1 6 6 . 9 7 3 7 7 , 1 2 2 0 7 . 6 H 6 S 6 , 0 7 2 9 6,0 7 6 6 8 , 9 0 0 o 9 , ! 0 5 0 bL 8 0 i s assumed that the pressure and density fluctuations i n the l i n e are small with respect to the mean values of pressure and density. Note that i n t h i s analysis the volumetric flow rate of the gas i s analagous to an e l e c t r i c current and the gas pressure to a voltage. The dis t r i b u t e d parameters which may then be derived for a l i n e of radius r (in.) are: ( 1 ) the dis t r i b u t e d resistance (Rp) : Rp = a * / * * * * - * * ] ^ * ( 5 . 1 4 ) where p- i s the viscosity (lb*sec/in 2) (2) the d i s t r i b u t e d inductance (Lp) : Lp = 8 * 1 t - i * r ~ 2 ( 5 . 1 5 ) where & =Po/g*R*T (lb*sec 2/in*) i s the mass density (3) the dis t r i b u t e d capacitance (Cp) : Cp = rrt.* r + 2*n-i*Po-i ( 5 . 1 6 ) Table 5 . 1 gives some calculated values for Rp,Lp,Cp. Using these variables, the admittance per unit length becomes: Yp = jwCp ( 5 . 1 7 ) while the impedance per unit length becomes: 81 Diameter (inches) 3/16 3/8 Rp ( l b - s e c 2 - i n - 6 ) 1.02x10-* 6.38x10-6 Lp ( l b - s e c 2 - i n - 6 ) 8 . 4 8 x 1 0 - * 2.12x10-6 Cp ( i n * - l b - i ) 7.67x10-* 3.07x10-3 Ambient pressure = 30 p s i . Temperature = 59°F Table 5.1. Some Values Of Hp, Lp, Cp Zp = Rp + jwLp (5.18) Consequently, the c h a r a c t e r i s t i c impedance (Zo) and propaga-tion c o e f i c i e n t ( TJ) may be written: Va. Zo = [Lp/Cp - jRp/wCp] (lb-sec-in - 5 ) (5.19) fc. [ (HP + jwLp) (jwCp) ] (in-i) (5.20) Using these variables, the transfer matrix of a section of pneumatic l i n e may be expressed as shown by Equation 5.21. P i I i cosh( L) Z o S l n h ( L ) 1 sinh( L) cosh( L) P T I T (5.21) 82 Figure 5 . 9 defines the variables used. Fig. 5.9. A Section Of Pneumatic Transmission Line Note that i t i s not possible to use the high frequency approximations for these transmission line equations. The reason for this i s that, at the frequencies of interest (f<20 Hz.) , the real-pacts of Zo and $ are not insignificant with respect to their reactive parts. Hence the complex hyperbolic functions in Equation 5 . 2 1 cannot be simplified. 5 . 1 . 3 Electro£neumatic Transducer The electropneumatic transducer i s the device which converts the e l e c t r i c a l signal of the feedback amplifier to a pressure signal for driving the bellows. A Fisher Type 5 4 6 transducer with pneumatic relay is used. The character-i s t i c s of this device are: Input Resistance 2 5 0 0 OLms Output Pressure Range 6 To 3 0 Psig. Damping Ratio .9 84 Natural Frequency 12 Hz. The l a s t two Figures were obtained by measuring the frequen-cy response of the transducer-relay combination with no output load; Figure 5.10 shows a Bode plot of the r e s u l t s . In the course of these measurements the importance of an adequate a i r supply became evident as ••clipping" of the output pressure signal resulted when the a i r supply could not provide the flew rate required by the transducer. For t h i s reason an accumulator (volume=3.5 cu.ft.) i s placed at the input to the transducer. 5.1.4 Tuning F o i l s The resonant frequency of the resonators i s changed by al t e r i n g the position of the tuning f o i l as shown in Figure 5.2. The res u l t i n g resonant frequency may be calculated from 1 ' ' ' i -1 +Af/fo = {1 + 2[ <wh>-<we>]/<w>} 2 (5.22) where: <wh*>,<we'> = change in the stored magnetic and elec-t r i c energy <w> = t o t a l stored energy i n the resonator *J . C. Slater, Microwave Transmission , McGraw-Hill,New York, 1942. 85 Since the tuning f o i l s are located at the root of the reson-ator, <we'>=0. The energy density per unit length at the root i s therefore ' where: jVo| =peak resonator voltage at the t i p s C = velocity of l i g h t Zo = c h a r a c t e r i s t i c impedance of the resonator Then the change i n the stored magnetic energy at the root for a displacement { AX) at one root piece (see Figure 5.2) <wh> = .5jVo|2*C-i*Zo-i (5.23) xs: <wh > = .5(8./2 + S t ) *| Vo|2*Ax/(C*Zo*40) (5.24) where: 5, = f r a c t i o n a l area of root piece occupied by upper and lower parts of tuning f o i l = .134 02= f r a c t i o n a l area of root piece occupied by central part of tuning f o i l = .140 The t o t a l energy stored i n the resonator i s <w> = L* | Vo I 2 / (4*C*Zo) (5.25) where L= resonator length = 121 i n . 86 Substituting 5.23 and 5.24 in 5.22 and simplifying gives Af = K f * ( A x 1 + Ax2 + Ax3 + Ax 4 + Ax5) (5.26) where Kf=15700 Hz./in. and the AX are defined in Figure 5.11. 5.2 Frequency Response Of The Pneumatic Tuning. System Figure 5.11 shows the organization of the system for which the vanalysis was done., Note that only one main air line (LS) per port was considered; therefore, the termina-ting impedance of this line was taken to be 1/2 that of the parallel arrangement of the 5 feeder lines shown (L1 to L5). Since a l l the pneumatic components in this system are coupled, the frequency response of the bellows, associated pneumatic lines and electropneumatic transducer must be cal-culated as one block. To do this, computer programs were written u t i l i z i n g the equations given in the previous sections—Figure 5.12 gives the flow chart of a program which finds the transfer function Af(jw)/ AV (jw) and the input conductance of the main line. The results were plotted on a line printer—Figures 5.13 to 5.15 show graphs for a 3/8" I.D. main air line. The following results were obtained: (1) The phase cross-over frequency of the system decreased with increasing line length. Table 5.3 shows some typ-i c a l values. Note that for short lengths this frequen-87 J-5_ ~ BELLOWS INPUT V PRESSURE JE5 BELLOWS (5) L4 P4 BELLOWS (4) L3 P3 FEEDER LINES PORT MAIN AIR LINE BELLOWS (3) L2 P2 BELLOWS (2) LI PI BELLOWS (I) 1/2 of JELLOWS _SERVED_ BY ONE PORT L6 ELECTRO-PNEUMATIC TRANSDUCER I Kf A f FEEDBACK SIGNAL (AV) F i g . 5 . 1 1 . Organization Of Pneumatic System Components Read i n Parameters Wri Parame te t ' ters i Calculate Rp, Lp, Cp for each 1i ne Find the input impedance of each feeder line at the port Find the terminating impedance of the main line Find the change In resonont frequency Af ((u) = .Kj.^ a X i (OJ) Find the pressure at the port '. . F i nd the i nput flow rate Find the input F i nd the input impedance of the pressure to the ma 1n linn ma i n line (E-P transducer F L O W C H A R T O F T H E C O M P U T E R P R O G R A M W H I C H 1 C A L C U L A T E S T H E F R E Q U E N C Y R E S P O N S E O F T H E * P N E U M A T I C S Y S T E M '< . Calculate ZO, GAMMA for each 1ine Find each be 1 lows di spl ace.r.-ent ; (tx i(',0> ' : .. ' : :r:":..!.: r- . . j ' Calculate the bellows input impedance Zb(W) Fi nd the p ress ure• at each be 11ows 1 ::: j , , : : 1 : \ : r : : ; ; : , . 1 1 : !. . . ! . . . . ; , , ! • : : 8 8 Fig. 5 . 1 2 . Computer Flow Chart 1": 3 9 T H E S E A R E T H E P A R A M E T E R S U S E D T O C A L C U L A T E T H E F R E Q U E N C Y R E S P O N S E O F T H E T U N I N G S Y S T E M P L A N T L E N G T H C L E N G T H ( L E N G T H ( L E N G T H ( L E N G T H C L E N G T H C L E N G T H ( F E E D E R 1 ) 2) 3 ) 4) 5) 6 ) 7) L I N E ( I N . ) ( I N , ) U N . ) ( I N , ) ( I N . ) ( I N , ) ( I N , ) 0 1 A ( I N ) F E E D E R L I N E : O I S T R E S I S T A N C E ( L B * S E C / I N * * 6 ' ) F E E D E R L I N E : O I S T I N D U C T A N C E C L B * S E C * * 2 / I N * * 6 ) F E E D E R L I N E : D I S T C A P A C I T A N C E C I N * * 4 / L B ) M A I N L I N E D I A C I N ) M A I N L I N E : D I S T R E S I S T A N C E C L B » S E C / I N * * 6 ) M A I N L I N E : D I S T I N D U C T A N C E C L B ^ S E C * * 2 / I N * * 6 ) M A I N L I N E : D I S T C A P A C I T A N C E C I N * * 4 / L B ) T E M P E R A T U R E ( D E G , F ) . V I S C O S I T Y ( L B * 3 E C / I N * * 2 ) P O L Y T R O P I C E X P O N E N T A M B I E N T P R E S S U R E C L B . / S Q , I N S ) E F F E C T I V E A R E A O F B E L L 0 * 8 ( S Q I N ) V O L U M E O F B E L L O W S C I N * * 3 ) B E L L O W S R E S O N A N T F R E Q , ( H Z ) B E L L O W S D A M P I N G R A T I O S P R I N G C O N S T , O F B E L L O W S C L B / I N ) D . C , E s P T R N S D C R R E S P . ( V O L T S * I N * * 2 / L B ) F - i P T R A N S D U C E R R E S O N A N T F R E Q , ( H Z ) E * P T R A N S D U C E R D A M P I N G R A T I O ( F R E Q , / B E L L O W S D I S P . ) C O E F , ( H Z / I N ) 63 95 127 159 191 1200 0 0 1,020600 8.48279D 7,66990D 0 6,37876D 2,120700 3.06796D 59 2,580Q0D 1 30 3 2 6 0 90 1 10 0 15700 ,00000 ,00000 ,00000 ,00000 .00000 ,00000 .0 ,18750 •-0 4 *0b «oa ,37500 • 06 •^ 06 -03 ,00000 *09 ,20000 ,00000 .60000 ,60000 ,00000 ,60000 ,00000 ,00000 ,00000 ,60000 ,00000 S Y M B O L S U S E D O N G R A P H ( M A I N L I N E L E N G T H V A R I E D ) 10 F T , 20 F T , 30 F T , 40 F T , 50 F T , 60 F T , 70 F T , 60 F T , 90 F T , 100 F T I S I S I S I S I S I S I S I S 4 I S Q I S # Table 5.2. Parameters For Fig. 5.13 To Fig . 5.15 7 5 , 0 0 9 0 , 0 0 9 7 , 5 0 m z o , i o i o 0,1cio 0 , 1 1 0 ? 0 , 1 2 ' 6 O . l I - i O 0 , 1 1 0 7 0 . 1 0 7 7 0 , 1 5 5 1 0 , It'll 0 , 1 7 1 0 0 , 1 7 0 6 I.IMl O,|0i>0 0 , 2 0 7 9 0 , 2 1 8 3 0,2?9J 0 . 2 O 9 7 0 , 2 5 2 7 0 , 2 6 5 5 0 , ? 7 * 6 0 , 2 9 2 5 0 . 3 0 7 2 0 , 3 ? 2 5 0 . 3 5 P 6 0 . 3 5 5 6 0 . 3 7 5 5 0 . J 9 2 0 0,0116 0 , 0 3 2 ? 0,0556 0 . O 7 6 S 0 , 5 0 3 5 0,5?S3 0 , 5 8 1 6 0 , 5 7 9 ? 0 , 6 0 - 1 0 , 6 3 5 5 0 , 6 7 0 5 0 . 7 C S 0 0 , 7 1 9 ? 0 , 7 7 6 2 0 , 6 1 5 0 0 , 6 5 5 7 0 , 6 9 6 5 0,91139 0 . 9 9 0 6 1 . 0 9 0 1 1 . 0 9 2 1 1 . H . 7 | , ? 0 a l 1 , 2 6 0 3 1 . 3 2 7 5 1 , 1 9 3 9 1 , 9 6 5 6 1 . 5 5 6 7 1 , 6 1 3 6 1,6511? 1 , 7 7 9 0 1,66(9 1 . 9 6 I J 2 , 0 5 9 0 2 , 1 6 2 3 2 . 2 7 C C 2 , 3 6 0 0 2 , 5 0 3 2 2,6?b5 2 , 7 5 9 7 2 . 6 9 7 7 3 , 0 0 2 6 3 , 1 9 0 7 5.3505 3 ,52?? 3 , 6 9 8 5 5,6632 0,0770 0,?M5 0 . 0 9 5 J 0 .7?0l 0,951.1 4 . 2 0 3 9 5 , 0 6 O | 5.7573 6 ,0?«l 6 , 3 2 5 0 6,6016 6 , 9 7 ) 7 7 , 3??0 7,6«H5 S , 0 7 ? 9 * 6,071)6 6.9C00 9,3050 06 160.00 3) PI O c m z o -< X N 16 9 3 Main Line 10 f t Length 100 f t . Main 3/4" 9.7 Hz 2.3 Hz Line 3/3" 6.8 Hz 2.4 Hz I.D. 3/T6" 5.0 Hz 2.6 Hz E-P Transducer not included table 5.3. System Phase Cross-Over Frequencies cy approaches the phase cross-over frequency of the bellows. (2) I f the pneumatic system i s approximated by a second-order system, an estimate may be made of the equivalent damping r a t i o s . Table 5.4 gives some t y p i c a l values f o r the system (the eff e c t of the electropneumatic transducer i s not included). Main l i n e length 10 f t 100 f t Main 3/4" .53 .26 Line 3/8" .46 .31 I.D. 3/16" .53 1.5 Ambient pressure = 30 p s i Temperature = 59°F Table 5.4. Pneumatic System Damping Ratios (3) An important factor which must be considered in the design of the tuning system i s the maximum value of the 94 main l i n e input conductance i n the frequency range of inter e s t . If t h i s value i s too h i g h — t h a t that i s , i f the flow rate required to produce a pressure sig n a l i s larger than the flow rate the electropneumatic transdu-cer i s capable of, then " c l i p p i n g " of the pressure w i l l r e s u l t . Table 5.5 gives some maximum values of the conductance (magnitude) f o r several l i n e parameters. These maxima occur at resonances i n the l i n e . Main Line Length 10 f t 100 f t Main 3/4" Line 3/8" I.D. 3/16" 338 at 10 Hz 124 at 4.5 Hz 54 at 2.55 Hz 1350 at 2.15 Hz 124 at 1.4 Hz 16 at 5.1 Hz Conductances given i n i n 3 / s e c - p s i Table 5.5. Maximum Input Conductances (1 cu. i n . per sec./psi = .035 cfm/psi) 5.3 Tuning System Construetion As a resu l t of the response ca l c u l a t i o n s i t was decided that a useful compromise i s achieved between the system's frequency response and i t s pneumatic input conductance by usinq a .375" I.D. main l i n e . To enable the electropneuma-t i c transducer to achieve the a i r flow rates required by th i s l i n e , an accumulator having a volume of 3.5 cu. f t . i s placed at the transducer's input. Because i t was found that the electropneumatic transducer was not adversely effected 95 by the magnetic f i e l d of the cyclotron provided the transdu-cer i s several feet away from the nearest pole t i p , the transducer-to-port (main line) length was reduced to f i v e feet. Figure 5 . 1 6 shows the organization of the tuning sys-tem. Simple i n t e g r a l control—implemented by the feedback amplifier—was found to be the best of several series feed-back compensation schemes. Figures 5 . 1 7 and 5 . 1 8 show the calculated system response that r e s u l t s when t h i s type of feedback i s used. in n (t) B (V r+ H-O o Ml t-3 C P H-0 vfl W w rt-(D B J f c J ACCUMULATOR ( 3 . 5 cu. f t * REGULATOR 1 8 psig setpoint BELLOWS PAIR L\V\/WNJ _ b J E-P TRANSDUCER TUNING 1 8 psig setpoint ( ± 1 2 psig signal) 3 / 8 " O.D. FLUX GUIDE PHASE PROBE FEEDBACK AMPLIFIER Main Air Supply: clean, dry ai r at 3 5 psig COUPLING LOOP r4-RESONATOR TRAN.-LINE PHASE PROBE PHASE DETECTOR •fl ua 3 ua P H* f+ C Cu (D O H i W (TJ W •tf o p w H p rt (D ua h n o P rt-ri O (0 • 40, i,10000E 00 0,1 1220 0,13589 0,11125 0, 15849 0,17783 0,19953 0,22387 0,25119 0,28184 0,31623 0,35481 0.398U 0,44668 0,50119 0,56234 0,63096 0,70794 0,79433 0,89125 1,0000 1,1220 1,2589 1,4125 1,5849 1,7783 ^ I, 9953 g 2,2387 a 2,5119 O 2,8184 ^ 3.1623 ^ 3,5481 s 3,9810 N 4,4668 5,0118 5,6234 6,3095 7,0794 7,9432 8,9125 9,9999 II, 220 12,589 1«,12S 15,849 17,783 19,952 22,387 25,119 28,184 31 ,622 35,481 39,810 44,668 50,118 56,234 63,095 '(0,794 79,432 89,124 • 30, Magnitude (dB) • •20, -10 I ! i I I 1 I 6 dB 22.5 dB" 0,0 98 X Ci Ci 1 / " • »• • 1 1 1 ' i M fr4 »•* — _ t 4^ *M M M • •-•»-» ^ 1 »>* »4 M 1 •*•» | "s. ^ 1 » ^ M 1 ! V . ; \ * * i >\ \ \ 1 1 1 I 1 0 I  r I \j 1 V 1 . 1 V I I ... . \ \ ! Frequency (Hz.) o o » m &• o f t j c o r j 3 © Oi i/J —« CO O - 3 LT CO ru i n to ru oo — -* *o a- =o co - m e ioo>? •O r*-> o *o 3 J 1 J i •a m a a w ru o r - a- —> rn o c* c* - C r - r - CO o o o o o o o o o o o o o o o o o o o o o r \ i ( o r u a i o i 7 i « ) ' - e o r u i i 3 ^ ^ ^ r o o * ^ i ^ w » A ; i O f u ? o o i n c o ^ i o r u o 0 ^ l \ 3 J i M M \ I ^ B - * i n f f 3 0 * n O ( M > 0 ' • > * » - t> m m • • m m m. m m m • « * • • • » • " • • • * • * * • • » • • •• • « ^ f t J ^ i n S v T r \ H / i ' 0 0 ^ l r H T 9 O ^ r t O l T ' 0 > Fig. 5.18. Phase Of Response, Integral Control 99 CHAPTER 6. RF FEEDBACK STABILIZATION Great care has been taken i n the design of the TRIUMF RF amplifiers to ensure that the lowest possible amount of noise i s introduced into t h i s system. However, by i t s e l f , t h i s w i l l not ensure that the RF tolerances demanded by the high energy resolution mode of operation (see Chapter 2) can be met. As an in d i c a t i o n of what i s required, i t must be noted that, without feedback, the noise l e v e l of the CRC1 RF system was -35 db when i t s driver was saturated and -30 db when i t s driver was not saturated. This performance l e f t SY NT HE SIZER PHASE AMP CONTROL CONTROL I 1 PHASE 23 MHz S H I F T E R A M P L I F I E R T R I P L E R 69 MHz A M P L I F I E R FINE TUNING RESONATORS AMPLITUDE PHASE CONTROL R E L A T I V E . 3 r d HARMONIC AMP & PHASE F i g . 6.1. RF Feedback Loops the 23 MHz CRC RF system far short of the -94 db noise l e v e l i TRIUMF Center Region Cyclotron. 100 desired. To enable the main cyclotron RF system to meet the desired performance requirements, the four RF feedback loops 1 i l l u s t r a t e d i n Figure 6.1 w i l l be employed. One may write the voltage that i s to be s t a b i l i z e d i n the following manner: Vr (t) =V1 {cos (w*t)-E*cos (3*w*t + P31) } (6.1) where V1 = fundamental (23.075 MHz) voltage amplitude E = amplitude r a t i o (third harm./fund.) V3 = E*V1 = t h i r d harmonic voltage amplitude P31 = phase of the th i r d harmonic with respect to the fundamental = P3~3*P1. This waveform i s s t a b i l i z e d by c o n t r o l l i n g the amplitudes (V1 and V3) and phases (P1 and P3) of the voltages generated by the two RF amplifier chains. With the exception of P1, the above variables are not measured d i r e c t l y . Instead, the variables measured are 2 Vp (peak RF voltage), E and P31. Using measurements of these variables to co n t r o l , respec-1 K.L. Erdman, K.H. Brackhaus, R.H.M. Gummer, Some Aspects of the Control and S t a b i l i z a t i o n of the RF Accelerating Voltage in the v TRIUMF Cyclotron, 6th Int. Cyclotron Conf., AIP Conf. Proc.#9 (AIP, New York, 1972) 444. 2 R.H.M. Gummer, Accurate Deterraination of the RF Waveform at TRIUMF , IEEE, Trans. Nucl. S c i . NS-18: No.3, 371-2, June 1971. 101 t i v e l y , V1,V3 and P3 couples these three feedback loops. To decouple these feedback loops, one must fi n d the values of V 1 , V3 and P3 i n terms of the measured quantities. This i s most e a s i l y done by l i n e a r i z i n g the relationships between the variables at the desired steady state. Two states of operation have been considered: (a) the high energy resolu-tion mode and (b) the high current mode, (a) hicjh energy resolution mode The steady state values of the waveform parameters for th i s case are: Vp = 100 kV. V1 = 112.5 kV. E = 1/9 = .1111 P31 = 0° Since V1 i s a function of the measured values {V1=V1 (Vp,E,P31)j, one may write dV1 = [|~]*dVp /[H--]*dE + [g-3T]*dP31 (6.2) where dV1, dVp, dE and dP31 are small changes in the varia-bles. The derivatives are evaluated at the operating point to give the following l i n e a r i z e d r e l a t i o n s h i p for d v l : dVl = ai*dVp + 12*dE + A3*dP31. (6.3) The peak voltage i n t h i s case i s 102 Vp = V1 (1-E) . (6.4) D i f f e r e n t i a t i n g then gives A1 = dV1/dVp = (1-E)-i (6.5) A2 = dVl/dE = V1/(1-E). (6.6) No simple formula exists which relates VI to P31. The value of A3 may be deduced, however, by considering the r e l a t i o n -ship of the fundamental and th i r d harmonic waveforms at the peak of the resonator waveform (see Figure 6.2). Because Vp coincides with a peak i n each waveform, the value of Vp i s not affected (to f i r s t order) by small changes i n P31. Therefore, A3=0, and the f i r s t of the required relationships may be written: (1~E)*dVl = dVp + Vl*dE (6.7) To relate the changes in the measured variables to changes in V3, one must d i f f e r e n t i a t e and l i n e a r i z e V3=E*V1. dV3 = E*dV1 + Vl*dE (6.8) substituting 6.7 i n 6.8 gives the required r e s u l t : 103 in CNJ in o '. — ' - 9 0 . 0 - 4 5 . 0 0 . 0 4 5 . 0 9 0 . 0 PHASE (DEG) F i g . 6.2. Resonator Waveforms (Maximum Energy Resolution) (1-E)*dV3 = E*dVp + V1*dE (6.9) The f i n a l variable which must be considered i s P3. The measured phase (P31) i s related to the difference between the phase s h i f t s of the two RF signals i n the following way: 1 0 4 P31 = P3 - 3*P1. (6.10) The change in the phase of the third harmonic may therefore be written dP3 = 3*dPl + dP31. - T V ' - . (6.11) (b) maximum current mode The steady state values of the waveform parameters for this case are: Vp = 100 kV. V1 - 102.5 kV. E = .24 P31 = -24° In this case the coefficients A1, A2 and A3 (Equation 6.3) are more d i f f i c u l t to find. To find A1, one must know the phase angle (Pp) at which the peak of the RF waveform may be found. H. Craddock* has derived these values; in this case Pp=-35.9°. One may then write Vp = V 1 {cos (Pp) -E*cos (3*Pp-P31) } (6.12) 1 M.K. Craddock, Effects of Third Harmonic on the SF Saveform (TRI-DN-72-15)# 1972, Fig. 4. 105 from which i t is evident that A1 = dVI/dVp = {cos(Pp)-E*cos (3*Pp+P31) }-i (6.13) No analytic expression was derived for A2 and A3; a numeri-cal value for these constants may easily be found however, from Figure 6.3.1 The values found were: A1 = 1.025 A2 = -66.7 kV, A3 r .35 kV./deg. And consequently dV1 becomes dV1 = 1.025*dVp - 66.7*dE + .35*dP31 (6.14) To find dV3, substitute the above Equation in Equation 6.8. This gives dV3= (E*A1) *dVp+ (E*A2 + V1) *dE+ (E*A3) *dP31 (6.15) dV3 = .25*dVp + 84.*dE +.08*dP31 (6.16) The relationship between the change i n the th i r d har-i H.K. Craddock, Fig. 5. 106 Fig. 6.3. Plots Of Constant P31 Vs VI And E (Vp=100 KV) 1 0 7 monic phase and the measured phases i s not altered by the mode of operation. Therefore i t i s s t i l l correct to write dP3 = 3*dPT+dP31 (6.17) In order to further analyse the operation of the RF feedback systems, the response of each amplifier chain to amplitude and phase modulation must be known. The method for calculating these envelope transfer functions i s derived i n Section 6.2. In addition to presenting the required transfer functions i t i s also shown that the coupling be-tween amplitude and phase modulation may be disregarded. When no ions are c i r c u l a t i n g i n the TRIUEF cyclotron, there i s no interaction i n the resonators between the s i g -nals at the fundamental and at the t h i r d harmonic frequen-cies . However, when ions are injected into the machine, the resulting c i r c u l a t i n g current provides a mechanism whereby the fundamental and t h i r d harmonic RF signals may i n t e r a c t . This interaction i s investigated i n Section 6.3 to determine the degree of coupling between the fundamental and the t h i r d harmonic parameters. 6.1 USSiva.lent C i r c u i t Of The TL-Resonator System In order to study i t s transient envelope response and the beam-RF interaction,,a lumped parameter equivalent of 108 the transmission line-resonator system1 was used. The equi-valent ci r c u i t is illustrated schematically in Figure 6.4. In -this figure the RF amplifier is represented by a voltage source (Vg) with an output impedance Rg. The resonant transmission line i s represented by the parameters R2, L2 and C2, while the resonators are represented by the parame-ters R1, L1 and C1. The coupling between the transmission line and the resonators i s represented by the mutual induc-tance (M). Although Figure.6.4 represents the transmission line and resonators using series resonant c i r c u i t s , parallel resonant circuits can equally well be used without affecting the inductive and capacitive values--only the values of the Fig. 6.4. Equivalent Circuit For TL-Resonator System 1 A. Prochazka, pp. 75-82. 1 0 9 r e s i s t i v e elements would be changed. The lumped parameters f o r the resonator are determined as follows: (1) given the energy stored i n the resonators (We) and the peak resonator voltage (Vr), the value of C1 i s then written Cl = 2*we/JVr|2 (6.18) (2) given the resonant frequency (wo, r a d . / s e c ) , the value of L1 then becomes L1 = ( w o 2*C1)-i (6.19) (3) given the unloaded Q of the resonators (QL), the value of E1 becomes H1 = ( Q L * W O * C 1 ) - I ( 6 . 2 0 ) The parameters representing the coupling and the resonant transmission l i n e can then be calculated i f the average mag-netic energy stored i n the l i n e <Wm> i s known, as well as the amplifier output voltage (Vg) and i t s current (Ig) . Given the quality factor of the l i n e (Ql) and assuming that 110 fo 23.1 MHz. Vr = 100 kV.. we = 23.7 joules QL 7169 Win = .115 joules QI 5593 ig = 223.7 amps Kg — 50.6 ohms Table 6.1. TL-Resonator Parameters (23.1 MHz.) Rg = 50.6 ohms R1 - 2.03x10-* ohms L1 = .01001x10- 6 henries C1 = 4740x10-12 farads R2 .238 ohms L2 = 9. 19x10--* henries C2 = 5. 17x10-12 farads M 6.97x10 -1° henries Table 6.2. Component Values For Figure 6.4 (23.1 MHz.) the l i n e , by i t s e l f , has the same resonant frequency as the resonators, one may then write the equivalent inductance f o r the l i n e as: 12 = 4<8m>/Ig2. (6.21) The lumped capacitance i s then C2 = (wo2*L2)~i (6.22) the series resistance becomes 111 Fo _ 69.3 MHz. Vr 13 kV. We = .425 joules QL 6209 Win = .0313 joules Ql = 4880 ig 45.65 amps Kg 62.54 ohms Table 6.3. TL-Resonator Parameters (69.3 MHz.) 1 Fo 69 .3 MHz. fig = 62 .54 ohms R1 7. 363x10-5ohms L1 1. 050x10~ 9henries C1 = 5. 0 2 4 x 1 0 - 9 f a r a d s R2 = 5. 362 ohms L2 = 6. 0 1x 10~ 5henries C2 = 8. 78x10-i*farads M 1. 49x 10-iohenries Table 6.4. Component Values For figur e 6.4 (69.3 MHz.) R2 = (wo*Ql*C2)-i (6.23) and the mutual inductance becomes M = [ (Rg-R2) R1 ]'/Z/wo (6.24) The requirement that the input impedance of the transmission line-resonator system match the output impedance of the 112 source (Rg=Vtube/Ig) at w=wo fixes the values of the equi-valent c i r c u i t representing the fundamental mode. Tables 6.1 and 6.3 give the parameters 1 required to calculate the component values for the fundamental and t h i r d harmonic equivalent c i r c u i t s . Tables 6.2 and 6.4 give the calculated d iscrete component values. From Table 6.1 one can see that, for the fundamental mode, more than 200 times as much energy i s stored i n the resonators as i s stored in the resonant transmission l i n e . This indicates that the time constant of the transmission line-resonator system i s la r g e l y determined by the time con-stant of the resonators. That th i s i s true has been v e r i f i e d 2 - - i t was found that t h i s system behaved l i k e a simple p a r a l l e l resonant c i r c u i t near the resonant frequency of the resonators with a bandwidth almost equal to the band-width of the resonators alone. The Q of t h i s system when driven by a properly matched source i s one half the Q of the system with the RF source removed. The introduction of beam current i n t o the cyclotron may be simulated by adding a current source (lb) to the equival-ent c i r c u i t of Figure 6.4 (see Figure 6.5). Depending on the phase r e l a t i o n s h i p between lb and the resonator voltage 1 A. Prochazka, pp.230-232. 2 A. Prochazka, communication. 3 A. Prochazka, p. 82 1 1 3 F i g - 6.5. TL-Resonator System (with Beam Current) ( V r ) , power may be taken from, or d e l i v e r e d to, the resonant c i r c u i t . 6.2 Envelope Response Trans.f_e_E F u n c t i o n s In order to i n v e s t i g a t e the t r a n s i e n t response of the RF system, i t was necessary to f i n d t r a n s f e r f u n c t i o n s which d e s c r i b e the output of a modulated c a r r i e r system, given the modulation of the i n p u t . F i g u r e 6.6 i n d i c a t e s s c h e m a t i c a l l y what i s r e q u i r e d — a t r a n s f e r f u n c t i o n which r e l a t e s the v e c t o r amplitude o f the output to the v e c t o r amplitude of the i n p u t . The amplitude i s w r i t t e n i n v e c t o r (complex) form because, i n g e n e r a l , the input to the c a r r i e r system w i l l be modulated both i n magnitude and i n phase. Thus a c a r r i e r modulated system may be most simply d e s c r i b e d by the v e c t o r which g i v e s the magnitude and phase of the c a r r i e r at a l l p o i n t s i n the system. The t i m e - v a r y i n g component of 1 1 4 modulated carrier system K t ) •Ira(Ai(t)*eJ w c t) H ( t ) » l m ( A o*eJ wc t) A l ( t ) Tm(s) s •55(t) equivalent amplitude transfer : function F i g . 6.5. E q u i v a l e n t Ampl i tude T r a n s f e r F u n c t i o n t h i s vec to r i s the modulat ing s i g n a l . For t h i s r e a s o n , the i n p u t s i g n a l i s w r i t t e n as I ( t ) = f ( t ) s i n ( w t + 0 ± ( t ) ) = I m ( f ( t ) e j 0 : L ( t ) e ; i W c t ) = Im(A7(t)ejWct) (6.25) where A i (t) i s the complex amp l i tude of the i n p u t s i g n a l . Lap l ace t r a n s f o r m i n g t h i s i n p u t s i g n a l y i e l d s I (s) =L { I m * e J W c * L - i ( H (s) } = ImAi (s-jwc) (6.26) (Note the use of the complex t r a n s l a t i o n theorem to get the f i n a l r e s u l t . ) The output of the system may then be w r i t t e n as H (s) = ImAi (s-jwc) *T (s) (6.27) 115 To remove the c a r r i e r from t h i s r e s u l t , one must transfer to the time domain; thus H (t) =Im £ e 3 W c t L - i [ al(s) T (s + jwc) ]} (6.28) H(t) = Im£e j W c t&o (s) ] (6.29) Comparing Equations 1.4 and 1.5 yields Ao(s) = Ax (s) *T (s + jwc) (6.30) Therefore the transfer function r e l a t i n g the input and output amplitude vectors i s Tm(s) •= T (s + jwc) (6.31) By considering the components of the amplitude, the transfer function i s changed to a transfer matrix having, however, only r e a l elements. Thus, i f the input and output are written as Ai(s) = xi{s)+jyi(s) (6.32) Ao(s) = xo (s) + jyo (s) ( 6 then the r e l a t i o n s h i p between input and output may be w ten in the following form: p -1 ReTm(s) -ImTm(s) x i ( s ) x ± ( s ) = TM y 0 ( s ) ImTm(s) ReTm(s) y±(s) y±(s) Figure 6 . 7 defines the variables used. Im Im Tm(s) Ai(t)| Re / |AD(t)| xi(t) x0(t) .Re Fig. 6 . 7 . I l l u s t r a t i o n Of Amplitude Vectors I f one i s only concerned with the changes in the am tude, then the input i s written as: A i ( s ) = r x s t + iVst-Ur Axi (s)+jAyi(s) ] (6 117 (xst, yst are constants) The output then becomes Io (s) =[*st J^iyst]*Tm (s) +[ Axi (s) + j Ayr (s) ]*Tm(s) (6.36) s The f i n a l value theorem shows that the f i r s t term on the right i s the steady state amplitude response. Thus one has — J 0 O Ao = R*e + [ Axi (s) + jAyi (s) ]*Tm (s) (6.37) or, in matrix form A x D ( s ) Axi(s) = TM» A y 0 ( s > A Y i ( s ) where TM was defined in Equation 6.31. 6.2.1 Amplitude, Phase And Frequency Modulation The vector amplitude transfer functions which were derived in the preceeding section have a limited usefulness since the quantities which are normally modulated are the amplitude, phase and frequency. Because the response of a linear system i s linear only for the vector components of the amplitude and not i t s magnitude and phase, linear trans-118 fer functions can be derived only for low levels of ampli-tude, phase or frequency modulation. To derive the required functions, the input to the system i s written as: — i A 0 i ( t ) Ai(t) = [ Ri+ARi (t) ]e (6.39) Note that, for simplicity's sake, the steady-state input phase has been chosen to be zero--this does not reduce the generality of the result since only a rotation of the co-ordinate frame i s involved. For small modulating signals one may then write: Ai(s) = L[ (Ri+ARi (t)) (1 + jA0i(t)) J (6.40) Ai (s) = Ri/s + ARi(s) + jRiA0i(s), (6.41) * v ' « A A i ( s ) Using Equation 6.38, the change in the output vector ampli-tude becomes Ao(s) =[ ARi(s)+jRi*A0i (s) ]T (s + jwc) (6.42) Figure 6.8 shows the output amplitude vector--the steady 119 J 0 O •is simply the steady state value J 0 O state component Ro*e j 0 o of (Ri/s) *T (s + jwc) —e>- Ri*Rg*e where Rg is the magnitude of T(jwc). From Figure 6.8 i t can be seen that the change Im(e-3 9 cAAo(s)) «^Re(e-J0<kAo(s)) =Aox Fig. 6.8. Output Amplitude Vector in the output magnitude ( ARo (s) ) it ARo=Aox=Re[ e~j0(&Ri (s) T (s+jwc) ] + Re[ je j 0°Ri (s) *£0i (s) T (s+jwc) J ( 6 . 4 3 ) The two terms of this equation give two transfer functions: (a) the amplitude-amplitude transfer function 1 ARo (s) /ARi (s) = Re[e~J0oT (s+jwc) ] (6.44) !p. R. Aigrain, B. R. Teare, E. M. Williams, Generalized Theory of  the Band-Pass Low-Pass Analogy, Proc. I.R.E. Vol. 3 7 , pp. 1152-1155; October, 1949. 1 2 0 = F^*Re[ T (s + jwc)/T (jwc) ] = Taa(s) (6.45) (b) the phase-amplitude transfer function -j0 n ARo (s)/A0i (s) = Ri*Re[je T(s+jwc)] (6.46) = -Ro*Im[ T (s+jwc)/T (jwc) ] = Tpa (s) (6.47) The change in the output phase ( A ^ O( S}) may also be derived by considering Figure 6.8; &0o(s) = tan~i[ Aoy/Ro ] = Aoy/Ro (6.48) Writing the last approximation out in f u l l gives AjzSo {s)=Im[e~^Rj£sVT (s + jwc) ] + Im[ je~J0£Ri_A0i(syr (s+jwc) ] (6.49) i 0 Ro from which the two remaining transfer functions may be ex-121 tracted. These are (a) the amplitude-phase transfer function A0o (s)/ARi (s) = Im[e (s + jwc) ]/Ro (6.50) = Im[ T (s+jwc)/T (jwc) j/Ri = Tap(s) (6.51) (b) the phase-phase transfer function -J0o Ao"o (s) /A^x (s) .= Im[je. T (s + jwc) ]*Rx/Ro (6.52) = Re[ T (s+jwc)/T (jwc) ] = Tpp (s) (6.53) Using the four transfer functions just derived, one may write a matrix equation relating the input and output modu-lation of a system. That i s , " A R 0 ( s ) " Taa(s) Tpa(s) ' AR ±(s)" A 0 o ( s ) -Tap(s) Tpp(s) 9 A 0 i ( s ) (6.54) The above results may be easily modified to analyse a 122 system's response to frequency modulation. This i s done by remembering that w = d0/dt which gives w (s) = s*&0{s) (6.55) Substituting the l a s t r e s u l t i n Eguation 6.54 re s u l t s in "AR 0(s)" Taa(s) lTpa(s) AR ±(s) = s 9 Aw Q(s) s Tap(s) Tpp(s) AW -L(S ) where £>wi(s) i s the input frequency modulation and A W O ( S ) i s the res u l t i n g output frequency modulation. 6.2.2 Calculation Of Modulation Transfer Functions The theory developed i n the preceeding sections has been used to derive the amplitude and phase modulation transfer functions for the TRIOMF RF system. 6.2.2.1 P a r a l l e l Resonant C i r c u i t A p a r a l l e l resonant c i r c u i t i s freguently used to pro-vide the impedance required by an RF amplifier stage. Figure 6.9 shows the equivalent c i r c u i t of such a stage and Equation 6.57 gives i t s transfer function. 1 2 3 K s 0 3s L + V ( s ) s C w2= 1 w c L F i g . 6 . 9 . P a r a l l e l Resonant c i r c u i t Vr (s) /I (s) — T (s) = R& QS 7 + S + Q (6.57) (8 =s/wc) The f i r s t step in deriving the modulation transfer functions i s to find the frequency translated transfer function T (s + jwc) (see Equation 6.31) where wc i s the c a r r i e r f r e -quency. Substituting (s+jwc) f o r (s) in 6.57 and s i m p l i -fying yields T(s + jwc) = R - j (R*Q/wc2) *_ 1 + r e n ( 6 . 5 3 ) taken to be equal to the resonant frequency of the c i r c u i t : wc= (L*C) . Since in an RF system the modulation frequen-c i e s are several orders of maqnitude smaller than the car-124 r i e r frequency, i t i s a very good approximation to write T (s + jwc) = R = Taa (s) . (6.59) 1 + r2Q_ls LwcJ Hence the transfer equation for the amplitude vector com-ponents i s A x D ( s ) Taa(s) 0 • A x ± ( s ) (6.60) A y D ( s ) 0 Taa(s) AYi(s) and that for amplitude and phase modulation i s AR D(s) Taa(s) 0 • ARi(s) (6.61) A 0 o ( s ) 0 Taa(s) R -J A 0 ± ( s ) Note that the l a s t equation shows that the coupling between amplitude and phase modulation i s n e g l i g i b l e . This w i l l be the case as long as 6.59 i s a good approximation—that i s , ReT (s+jwc) >>XmT (s + jwc) . 125 6.2.2.2 2 Cou^ed, Resonant Circuits The circuit shown in Figure 6 . 1 0 describes the last stage of the TSIUflF RF amp.lil.fier chain (the Transmission Line-Resonator System). Note that there are two sources F i g . 6.10. Equivalent C i r c u i t For The TL-Resonator System contributing to the resonator voltage (Vr ( s ) ) — t h e output voltage (Vg (s)) of the f i n a l tube i n the RF chain and the cyclotron beam current lb (s). Hence two transfer functions T (s) and Z(s) must be used to r e l a t e the resonator voltage to the inputs. Equation 6.62 shows this r e l a t i o n s h i p Vr(s) = T (s) Vg (s) + Z (s)'lb (s) (6.62) where the transfer functions, written i n terms of the norma-l i z e d variable S (6 =s/wc) are: (6.63) 126 • • • • , • ' ( 6 . 6 4 ) Z(s) = 1 - Ql (Q"?fc2+o+Q2)  Sw cCi (iWcCi)((Qlg2+S+Ql)(Q262+8+Q2) - S ^ Q i Q 2 ) The parameters used in the l a s t two equations are defined i n terms of the system's components (Figure 6.10) as: W C2 = (C1*L1)-*= (C2*L2)~i k = M/ (L 1*L2),/7-Q1 = wc*L1/R1 Q2 = wc*L2/(R2 + Rg) Ao = k*Q1*Q2/[ wc2 (L1*L2/2C1 ] To fi n d the amplitude response at the resonant freguency, S i s set equal to j , giving T(jwc) = Ap ( 6 . 6 5 ) 1 + k^Q]:Q2 (ie 8 = s -» j*w = j for w = wc ) wc w„ Z(jwc) - (Ql/wcC1) . r , 1 _ _ - i _ l " L 1 + k2Q1Q2 Ql J (6.66) 1 2 7 To fi n d the frequency translated functions, substitute 6 +j for & i n 6.63 and 6.64. (This assumes that the c a r r i e r frequency i s equal to the resonant frequency (wc).) Since (s) i s now c h a r a c t e r i s t i c of the modulating frequencies, l 8 l=|s/wc|<<1, and hence i t i s a very qood approximation to write , T (s + jwc) = Ao (28j-1) /(Ex + jEy) (6.67) Z (s+jwc) = r Ql 1*f2Qg& + 1 - j _ l _ ] (6.68) w cCi E x Ql where: Ex = (2*Q1»S +1) (2*Q2*S +1) +k2QlQ2 (6.69) Ey = St (Q^ S+1) (2*Q2*S+1) + (Q2*S+1) (2*Q1*&+1) +4k*Q 1Q2 j (6.70) Examining the dominant terms of these l a s t equations i n d i -cates that |'Ey/ExJ^l S|. This permits the s i m p l i f i c a t i o n of 6.68. Taking the r e a l and imaginary parts of the s i m p l i f i e d 6.67 and 6.68 gives: 128 ReT (s + jwc) = AQ ( 6 . 7 1 ) (1+Tis)(1+T2s) + k^QiQ2 _ r o i o z i s [ r_^f s2+ s roi±Q21 _ k 2 ImT (s + jwc) = I J U w J S L OjOoJ K J (6.72) ((1+T 1 S)(1+T2S) + k2Q 1Q 2)2 ReZ (s+iwc) = Qi * 1 + Tos (6.73) w cCl (1+Tis)(1+T 2s)+ k2QiQ 2 ImZ (s + jwc) = - ( W C * C 1 ) - i ( 6 . 7 4 ) (T1=2*Q1/WC, T2=2*Q2/wc) Using the above functions then enables one to write the matrix equation which r e l a t e s the change in the output vector amplitude to the changes in the input vector- ampli-tudes. That i s , 129 AVrx(s)[ AVry(s) + ReT(s+jw c) -ImT(s+jw c) * AVgx(s) ImT(s+jwc) ReT(s+jw c) AVgy(s) "ReZ(s+jwc) -ImZ(s+jw c) * 'AIbx(s) ImZ (s+jw c) ReZ(s+jw c) Alby(s) (6.75) F u r t h e r s i m p l i f i c a t i o n i s p o s s i b l e . Since the imaginary p a r t s of 6.67 and 6.68 are very s m a l l with r e s p e c t to the r e a l p a r t s , the o f f - d i a g o n a l elements of the l a s t equation may be set to zero. T h i s g i v e s : r_oi l AVrx is) ~ AoAVgx(s)+LwcClJ»(l+T2s)AIbx(s) (6.76) (1+Tis)(1+T2s) + k2QlQ2 r_9jq AVrv(s,= AQAVgy(s) +Lw cC 1>(l+T 2s)AIby(s) (6.77) (1+T 1 S)(1+T 2s) + k2Q!Q2 Using 6.71 and 6.73 and the same assumption as above l e a d s t o an amplitude response which may be w r i t t e n as AAvr(s) = A c yAAvg(s) +[wcCi"}(l+T2)AAib(s) (6.78) (l+T l S)(1+T 2s) +k2Q!Q2 and a phase response which may be w r i t t e n as A O v r = A0vg(s) + (l+k 2QiQ 2)(l+T 2s)A01b(s) (l+Tis)(1+T2s) + k 2 Q l Q 2 (6.79) 1 3 0 The terms coupling the amplitude and phase response have been neglected because, as has been already noted, the ima-ginary parts of T (s+jwc) and Z (s + jwc) are negligible with respect to their real parts. From the last three Equations one can see that the characteristic equation of the TL-Resonator envelope res-ponse is D(s) = ( 1 + T1*s) (1 + T2*s)+k2QlQ2. (6.80) a more useful form of this equation is D(s) = T1T2[ s2+s* (T1+T2)/(T1T2) + (1 + b) .] (6.81) where b=k2QlQ2. The roots of this quadratic (and the poles of the TL-Resonator system) are: s = [-1± (1-4T1T2 (1 + b)/(T1+T2) 2)^ ][ T 1+T2 ]/[ 2T1 T2 J (6.8 2) Because T1>>T2 for the TL-Resonator system, one can simplify the above expression by using the following approximation: [ 1-4T1T2 (1 + b)/(T1+T2) 2 ] = 1-2 (1+ b) T2/T1. (6.83) using this approximaton, the new roots of the characteristic polynomial become 131 Fundamental T1 = 98.9x10-6 sec. T2 = .361x10-6 sec. Ta(exact) = 49.555x10-Tb (exact) = .362x10-6 6 sec. sec. Ta(approx.) = 49.6x10-- Tb(approx.) = .361x10-6 sec. 6 sec. Third Harmonic T1 = 28.52x10-6 sec T2 = 1.77x10-6 sec. Ta (exact) = 14.56x10-6 Tb (exact) •= 1.882x10-* sec. sec. Ta(approx.) = 15.48x10 Tb(approx.). = 1.77x10-- 6 sec. 6 sec. Table 6.5. Time Constants For The TL-Resonator System s1 = (1+b) (T1 + T2)/T1* ~ - (1+b)/T1 (6.84) s2 = [-1+(1+b)T2/T1][T1 + T2]/[TlT2] = -T2~* (6.85) Equation 6.81 may therefore be rewritten as D(s) = (1+k*QlQ2) (1+Ta*s) (1+Tb*s) (6.86) where Ta=Tl/(1+.b) and Tb=T2. Using the above re s u l t to re-write the transfer functions for the vector components of the c a r r i e r modulation gives 132 AVrx(s) = A T A V R X C S ) + A ? ( l + T 2 s)AIbx(s) ( 6 . 8 7 ) ( 1 + T as) ( 1 + T bs) AVry(s) = AiAVgy(s) + A ? ( l + T 2 s)AIby(s) (6.88) ( 1 + T a s ) ( l + T bs) where A1 = Ao/[ 1+k2QlQ2 ] (6.89) A2 = Rl*Q1 2/[ 1+k2QlQ2]. (6.90) Table 6.5 l i s t s the time constants for both the fundamental and the third harmonic mode. 6.3 The RF--Beam Interaction The transfer function r e l a t i n g the c i r c u l a t i n g beam current (lb) to the resonator voltage was derived i n Section 6.2.2. However, the magnitude of the beam current (and the magnitude of i t s fundamental and t h i r d harmonic components) in turn depends on the resonator voltage. Also, the funda-mental and t h i r d harmonic RF systems i n t e r a c t , respectively, with the fundamental and t h i r d harmonic Fourier components of the beam current pulses. Since any change i n the shape 1 3 3 of these current pulses affects both their fundamental and their third harmonic components, there will be an interac-tion between the two RF systems, the strength of which must be investigated. Under the expected steady state operating conditions 1, the fundamental RF system transfers power to the beam. However, the third harmonic system (due to the relationship between the third harmonic RF voltage and the third harmonic component of the beam current) draws power from the beam. 6.3.1 Instantaneous Circulating Current lTheory}_ In order to calculate the instantaneous circulating current in the TRIUMF cyclotron, i t i s f i r s t necessary to know the transit time (ta) of an ion leaving the machine at a time t. That i s , a particle leaving the machine at time t will have been injected into the machine at an earlier time (t - ta). Because the ions are injected over a range of phases (p) with respect to the RF fundamental peak, the i n -stantaneous transit time i s a function of both time and phase. ta = ta (p,t) (6.91) 1 A. Prochazka, K.I. Erdman, Estimate of Beam Loading, TSI-DN-71-39, October 1971. 1 3 4 Figure 6 . 1 1 shows the injected beam current pulse for the maximum energy resolution mode and illustrates the relation-ship between this pulse and the components of the RF voltage waveform. 0 . 0 9 0 . 0 Fig. 6 . 1 1 . RF Voltage S Injected Current Waveforms - 2 7 0 . 0 - 1 8 0 . 0 - 9 0 . 0 P H A S E (DEGJ Assuming that a l l ions leave the cyclotron with the 1 3 5 same f i n a l energy (Ef/eV), then the time an ion spends i n the machine depends on i t s rate of energy gain. This rate may be written dE(p,t)/dt = 4*q*Vr (p,t) * f c (6.92) where E(p,t) = ion energy (eV) q = ion charge (e -) Vr(p,t) = fiF resonator voltage fc = ion rotation frequency (Hz.) 2*q*Vr(p,t) = energy gain per gap crossing. Integrating the l a s t Equation gives Vr(p,x)dx = Ef/4*q*fc . (6.93) J t-t a(p,t) This i s the equation which must be solved for ta (assuming Vr(p,t) i s given). Because the time between input current pulses i s ( 5 * f c ) _ 1 , the number of current pulses injected into the machine during the time ta i s Np(p,t) = 5*fc*ta (p,t) . (6.94) One f i f t h of these c i r c u l a t i n g current pulses cross the ac-celerating gap every half SF cycle. Therefore the current crossing the accelerating gap i s given by Np Ib(p,t) = .2* J ^ I i n (p,t) (6.95) 1 36 where I i n i s the current injected into the machine as a function of time and RF phase. That t h i s summation may be approximated by an in t e g r a l i s apparent from Figure 6.12, t I i n F i g . 6.12. Integral Approximation For C i r c u l a t i n g Beam Current from which one can deduce that Np E n= 0 A t * I i n (p,n) ) cf I i n (p, x) dx. to (6.96) Because At<<ta, i t i s a good approximation to write Ib( P,t) = 1 * 1 5 At, I i n (p,x) dx t-t a(p,t) (6.97) where At=1/5*fc. If Iin(p,t) i s constant with respect to 137 time (the usual operating condition) then Ib(p,t) = fc*Iin (p) *ta (p, t) . (6.98) Note that, because the resonator voltage varies with time, ta is also a function of time. To find the rate of change of the transit time, one may differentiate Equation 6.93, Vr(p,x)dx) = 0 (6.99) t-tf p , i ' giving Vr(p,t) - [ Vr (p,t-ta) ][ 1 - dta/dt] = 0. (6.100) The required differential equation for the transit time may then be written: dta/dt = 1 - Vr (p,t)/Vr (p,t-ta) (6.101) No analytic solution exists for this equation—it i s there-fore not possible to find ta(p,t) by integrating Equation 6.101. One may, however, find ta numerically by making use of Equation 6.93. If a record i s made of the integral of the resonator voltage [ J v r ( p , t ) d t = IVr(p,t)] then IVr(p,-ta) = IVr(p,0) - Ef/4*q*fc (6. 102) 1 3 8 4 IVr(p,t) Fig. 6.13. Transit Time From The Integrated Resonator Voltage where t was set equal to zero. As i s seen from Figure 6.13, ta may be found ty searching backwards from t=0 for the value of IVr which satisfies the last equation. The beam current may then by found by using Equation 6.97, or, i f the input current i s constant. Equation 6.98. To find the rate of change of the beam current, one must differentiate Equation 6.97 with respect to time. This gives dlb(p.t) = (Iin(p.t) - Iin(p,t-t a)*(i - dt a))f c (6.103) dt dt or, i f the input current i s constant with respect to time, d[ lb (p,t) ]/dt = fc*Iin (p) *[ dta/dt ]. (6. 104) 139 6.3.2 Beam Current Calculations ' 6.3.2.1 Rectangular Current Pulses If the phase width (Pw) of the input current pulses (Figure 6.11) i s small, then the shape of the c i r c u l a t i n g current pulses w i l l remain approximately rectangular. The Fourier components (fundamental and t h i r d harmonic) of these Fundamental RF Only lav 100x10-6 amps Pw = 45° PI = 0°±15° i n i t i a l l y Max. Energy Resolution lav = 45x10-6 amps Pw = 20°±5° PI = 0° Max. Current Density lav = 133x10-6 amps Pw = 60° PI = 0° Table 6.6. Input Beam Current Parameters pulses may then be used to estimate the strength of the RF— beam int e r a c t i o n . To simplify the Fourier analysis of the beam current pulses, translate the o r i g i n to a point half-way between the 140 pulses (p=p° + 90°+Pl°-) . Because the current waveshape i s an odd valued function with respect to t h i s new o r i g i n , the Fourier series representing the pulses w i l l consist of only sine terms. That i s Ib(p) = ^ B n * s i n ( n * p ) (6.105) n=l where B n = 1 . ntCJ lb (x) * s i n (n*x) dx . (6.106) Evaluating the above i n t e g r a l gives Bn =(-1J *4*lp*sinr f2m-11Pw/2j (6.107) (2m - l)u (n = 2*m-1, m=1,2,..) and the current waveform may be written 00 Ih(p) = 4»Ip T i l ) ( n»sin[ (2n-1) Pw/9]*sin[ (?n-1)p] (6.108) K. n=l ( 2 n _ 1 ),. from which one can see that only the odd harmonics are i n -cluded in the se r i e s . Translating back to the o r i g i n a l phase o r i g i n transforms the Fourier expansion to Ib(p) = 4Ip_y_l sin((2n-l)Pw) cos ( (2n-l) (p+pi) ) (6.109) * ^ n - l 2 141 the fundamental component of the beam current pulses i s therefore Ib1(p) = (4 *1?^ ) *sin (Pw/2) *cos (p+Pl) (6.110) and the t h i r d harmonic component i s Ib3(p) = (4*Ip/3«) *sin (3Pw/2) *cos (3*p + 3*Pl) . (6.111) the amount of current injected into the cyclotron i s meas-ured by i t s average (lav) over an EF cycle. Given the phase width of the pulses (Pw°) and the t r a n s i t time (ta), the peak value of a rectangular pulse of c i r c u l a t i n g current may then be written: l p = fc*ta*Iav*360./Pw (6.112) where f c = the ion rotation frequency. Given the amplitude of the fundamental component of the rectangular current pulse, the t h i r d harmonic amplitude can be written |Ib3| = [Ib1][1 + 2*cos(Pw)]/3 (6.113) 142 6.3.2.2 Current guises Shaped By. Resonator Voltage I t i s assumed that the input current [Iin(p,t) ] varies with RF phase as shown i n Figure 6.11. It i s also assumed that t h i s input does not vary with time, so that the c i r c u -l a t i n g beam current becomes Ib(p,t) = f c * I i n (p) *ta (p,t) (6,114) To f i n d t a , Equation 6.93 must be used. 11L invariant resonator voltage Vr(p,t) = Vo[cos(p) - E*cos (3*p+P3) ] = Vr (p) (6.115) In t h i s case Equation 6.93 becomes ta(p)*Vr(p) = Ef/4*g*fc (6.116) Substituting ta from the l a s t equation into Equation 6.98 shows that the waveshape of the c i r c u l a t i n g beam current i s Ib(p) = Ef*Iln ( N -Pw/2 + PI < p < Pw/2 + PI (6.117) 4qVr(p) --= 0 elsewhere. From t h i s l a s t equation one can see that Ib(p)*Vr(p) •= con-143 stant; that i s , in the steady state the instantaneous power delivered to the beam i s constant throughout the phase width (Pw) of the beam current pulses. Since a rectangular input current pulse i s assumed, the peak of t h i s waveform (Iin) may be found from the average input current: I i n = Iav*360°/Pw° (6.118) Using the input current parameters from Table 6.6 and the max. beam current mode -90.0 Fig. 6.14. -45.0 0.0 P H A S E IDEG) 45.0 90.0 C i r c u l a t i n g Beam Current Waveforms 144 RF voltage parameters from Table 2.2, the c i r c u l a t i n g cur-rent waveforms were plotted for both the maximum energy re-solution mode and the maximum current density mode (see Fundamental only _£_1 OOkV £eak]__ Ib1 = .50 amps Pb1 = 0° Ib3 = .21 amps Pb3 = 0° lax. Energy Resolution Ib1 = .22 amps Pb1 .= 0° Ib3 = .21 amps Pb3 = Oo Max. Current Density Ib1 = .77 amps Pb1 = -1.4° Ib3 = .52 amps Pb3 = -5.1° where: Ib1 = amplitude of the fundamental component Pb1 = phase of the fund. comp. w.r.t. the fund. RF peak Ib3 = amplitude of the t h i r d harm, component Pb3 = phase of the t h i r d harm. comp. w.r.t. the fund. RF peak Table 6.7. C i r c u l a t i n g Beam Current (Fourier Components) 1 G. Dutto (private communication). 145 F i g . 6.15. Beam Current Components Figure 6.14). These waveforms were Fourier analysed using a computer program—the r e s u l t s of these c a l c u l a t i o n s are d i s -played i n Table 6.7. Note that the phase i s given with res-pect to the peak of the reference RF fundamental voltage. Figure 6.15 shows plots of the Fourier beam current com-ponents l i s t e d in Table 6.7. 146 i l l Time Varying Resonator Voltage The simplest case was considered—that of an amplitude modulated resonator voltage consisting of only the fundamen-t a l frequency. In t h i s case the resonator voltage i s Vr(p,t) = [ Vo*cos (p) ][ 1 + m*cos(wm*t)] (6.119) where m = modulation r a t i o wm •= modulation frequency (rad./sec.) Substituting the above voltage i n Equation 6.93 and integra-t i n g gives the r e s u l t : w m t a + m*sin(w mt a) - m*sin(w m(t-t a)) = Efw m (6.120) 4qf cVoCOs(p) This equation must be solved for t a ( p , t ) . A general solu-tion i s unavailable—however, approximate solutions e x i s t f o r low and high modulating frequencies. By low modulating frequencies i t i s meant that wm*ta<<1. Expanding the l a s t equation gives w m t a + m*sin(w mt) - m*sin(w mt)cos(wmta) (6.121) + m*cos(w mt)sin(w mt a) = w mEff 147 where Ef f = Ef /4*q*fc*Vo*cos (p) . Applying the low frequency approximation y i e l d s ta(p,t) •= Ef/4*q*fc*Vr (p,t) (6.122) and the c i r c u l a t i n g beam current becomes Ib(p,t) = Iin{p)*ta(p,t) (6-12.3) As for the stationary case, lb (p,t)*Vr(p,t) i s a constant. The foregoing has actually been a restatement of what i s i n t u i t i v e l y obvious--that i f the resonator voltage changes very slowly with respect to the tr a n s i t time, then the i n s -tantanteous current waveshape may be calculated from the instantaneous accelerating voltage waveshape. The high frequency approximation for ta i s obtained by considering modulating frequencies such that wm*ta>>1. I f t h i s condition i s s a t i s f i e d , then the term wm*ta w i l l dominate the l e f t hand side of Equation 6.120. The t r a n s i t time then becomes ta(p,t) = Ef/4*q*fc*Vo*cos (p) = ta (p) (6.124) and the c i r c u l a t i n g beam current becomes Ib(p,t) = Iin*Ef/4*g*fc*Vo*cos (p) (6.125) 148 In t h i s case the instantaneous power delivered to the beam at a pa r t i c u l a r phase i s not constant; however, the average power i s . Since the t r a n s i t time for a 500 MeV beam i s ap-proximately .25x10-3 s e c , one may conclude that the wave-shape of l b w i l l not be s i g n i f i c a n t l y affected by resonator voltage deviations having frequencies large with respect to 4 kHz. (=1/ta). A comparison of the Fourier beam current components given in Table 6.7 and the voltage waveform components l i s t e d in Table 2.1 and 2.2 shows that the phase r e l a t i o n -ships between the voltage and current are such that power i s fed from the resonators to the beam in the fundamental mode and from the beam to the resonators i n the third harmonic mode. Should the beam feed more power into the t h i r d har-monic resonator mode than i s required to maintain the de-sired third harmonic voltage, then a means must be provided to dissipate the excess power (lower the t h i r d harmonic Q). This i s not necessary however, because the energy d i s -sipated i n the resonators i n the t h i r d harmonic mode far exceeds the power fed into t h i s mode by the beam. For exam-ple, one f i n d s that 27.3 kW are dissipated by the t h i r d har-monic i n the maximum energy resolution mode while 2.6 kW are supplied by the beam. S i m i l a r l y , for the maximum current mode, the power dissipated i s 105.9 kW and the power sup-plied i s 12.7 k». 149 6.4 Frequency Response Calculations The i n i t i a l investiqations of the RF amplitude and phase control systems were made using frequency response methods. In these calculations the ef f e c t of coupling be-tween these systems was disregarded. The design produced i n t h i s way was then checked by simulating the entire RF con-t r o l sysem, coupling effects included. 6.4.1 Fundamental RF System Because t h i s study was done before the transmission line-resonator system was assembled, the transfer functions for this component were calculated using the formulas derived in Section 6.2. The amplitude response transfer function was found to be Ta(s) = Ar (6.126) (1 + T a s ) ( l + T bs) while the phase response function was found to be Tp(s) •= 1 . (6.127) (1 + T a s ) ( l + T bs) The l a s t two equations were derived from Equations 6.78 and 6.79 by using only the f i r s t term i n t h e i r numerators (eg. disregarding the term due to beam loading). The denominator, given by Equation 6.86, has time constants Ta = 49.56x10-* seconds and Tb = .362x10~6 seconds. (a) 1 usee./div., lower trace is the input amplitude modulation, 1.5% modulation ratio. (b) 10 ysec./div., lower trace i s the input amplitude modulation, 7% modulation ratio. (c) 20 ysec./div., lower trace i s the input amplitude modulation, 1.5% modulation ratio. Fig. 6.16. Fundamental RF System Step Response. 1 5 1 The transfer functions for the remaining portion of the fundamental RF amplifier chain (including detectors) were obtained by observing the output modulation when the driving s i g n a l f or thi s system was amplitude modulated. For these measurements the power generated by the RF system (about 1 Ww). was dissipated i n a dummy l o a d 1 . Figure 6.16 shows the response of the RF chain to step modulation of the input amplitude--this response was approximated by the function TRF1(s) A*e~ s T l # (6.128) [s_] ^  + [2_] s + 1 where TI = time lag | = damping factor wn = undamped natural frequency (rad/sec). From the photographs (Figure 6.16(a)) the time lag (TI) i s seen to be 1.3x10~6 seconds. To f i n d g and wn, the plot shown i n Figure 6.17 was used. From the measured per-unit overshoot of the response (u=| (YO-Y1)/YOj), the damping factor was found by using Equation 6.129. f ln(u) < 6 ' 1 2 9 ) (^2 + ( l n ( u ) ) 2 ) % 1 This dummy load i s a c i r c u l a t i n g soda-water device b u i l t by Brown Boveri Ltd. 152 A t Fig. 6.17. Step Response (2nd Order System) The undamped natural frequency was then obtained from wn = (6.130) tp(l - S 2)^ where tp is defined in Figure 6.17. From the photographs shown, i t was found that | =.6 and wn = 13.8x10+5 rad/sec. 153 6.4.1.1 fundamental Amplitude Regulation The design of t h i s system was accomplished by adding series compensation to the feedback loop to get a suitable closed loop response c h a r a c t e r i s t i c 1 ^ 3 , * . Figure 6.18 shows a block diagram of t h i s system. I t was found that a simple lag compensator with time constants T1 = Ta = 50x10- 6 sec-onds and T2 = 1.6x10 - 3 seconds s a t i s f i e d the design c r i -t e r i a . That i s , i t was found that with t h i s compensator the system had a large open loop gain at low frequencies (54 db at 100 Hz.) and a high gain cross-over freguency (47.9 kHz.). A gain margin of 6 db and a phase margin of 42° were chosen to give a suitably stable response. Figure 6.19 shows the magnitude and phase response of the amplitude con-t r o l system components, as well as the response of the open and closed feedback loop. 1 K.H. Brackhaus, Preliminary Calculations for the CRM Amplitude Control System, TRI-DN-71-35, September 1971. 2 K.H. Brackhaus, Design of a Stable 1.50 KW 23 MHz Amplifier for the TRIUMF CRM, Chapter 3, 1972. 3 B.O. Watkins, Introduction to Control Systems, pp 382,441. * Y. Takahashi, H. Robins, D. Auslander, Control and Dynamic Systems, pp 356,398. 1 5 4 (1+Tas) (1+Tbs) RF Chain TL-Res Syst F i g . 6.18. Block Diagram Of Amp. Control System 6.4.1.2 Fundamental Phase Regulation This system d i f f e r s from the fundamental amplitude re-gulation system i n only one r e s p e c t — a high pass element with a break freguency of 3Hz. was added to t h i s system, thus permitting the phase of the fundamental resonator RF voltage to d r i f t with respect to that of the synthesizer reference. This means that a very stable phase reference i s not needed^ thereby simplifying the control c i r c u i t y . Con-t r o l of the fundamental RF phase i s not required f o r low frequency errors (see Figures 2.8,2.9) because the beam i n -jec t i o n system i s locked to the fundamental RF phase. 156 F i g . 6.20.- Amp Response (Fund Phase Control System) 157 F i g . 6.21. Phase Response (Fund Phase Control System) The seri e s compensator chosen for t h i s c o n t r o l systam i s given by Tcomp(s) = s*(1 + T1*S)/(1 + T2*s) 2 (6.131) where T1 = 50x10~ 6 seconds and T2 = .05 seconds. Figure 6.20 shows the amplitude response of the phase con 1 5 8 t r o l system and Figure 6.21 shows the phase of the response. As i s to be expected, t h i s system's gain margin, phase mar-gin, gain crossover frequency and phase crossover frequency are nearly i d e n t i c a l to that of the fundamental amplitude control system. 6.4.2 Third Harmonic RF System The transfer functions for the transmission line-reson-ator system were derived using the Equations given i n 1 + 4.6x10"7s res (1+1.5x10 ~ 5s)(1+I.9xl0 6s) RF Syst Compensator TL-Res Syst Detector 1 + 10"5s 1 + 10 " 6s J Ref 1 + 3.2xl0" 5s Fig. 6.22. Third Harmonic F-B Loop (Amp Control) Section 6.2. Using these equations i t was found that the amplitude response could be represented by Ta (s) = Ar (1 + T a s ) ( l + T bs) (6. 132) and the phase response by Tp(s) = 1 (1 + T a s ) ( l + T bs) (6. 133) 159 where Ta = 14.56x10~6 seconds Tb = 1. 882x10~6 seconds. The transfer function for the third harmonic RP chain (excluding the f i n a l stage) was estimated from the c i r c u i t shown in Figure 1.2. Since this amplifier was s t i l l in i t s preliminary design stage, the Q of the two driver stages was not known. For the purposes of this design study, a Q of 100 was assigned to each. Using Equation 6.59, this gave TRF3(s) = A (6.134) (1 + T*s)^ for the required transfer function where T = Q/ (TC *f) =100/ (it *69* 10*) =. 46x 1 0~& seconds and A = 350 for the amplitude response A = 1 for the phase response. The detectors which will be used with the third har-monic feedback controls are sufficiently slow that their response must be taken into account. Two detectors are involved 1: (a) a third harmonic ratio detector which finds the ratio of the third harmonic amplitude to the fundamen-tal amplitude 1 R.H.M. Gummer, Accurate Determination of the RF Waveform at TRIUMF , IEEE Trans. Nuc. Sci. NS-18: No.3, 371-2, June 1971. 1 6 0 and (b) a third harmonic phase detector which finds the phase of the third harmonic peak with respect to the fundamental peak. Both these detectors use for their input a 10 kHz. replica of the resonator waveform that is produced by a synchronous sampling technique. The frequency response of these devices consists essentially of a simple lag with a break frequency that must be small with respect to 10 kHz. Figure 6.22 shows a block diagram of the third harmonic amplitude con-tr o l feedback loop. A simple lead-lag series compensator was found to be adequate to ensure a stable response with a large open loop gain. For a larger loop gain at low frequencies the break frequency of the detector can be decreased from 5 kHz. to a suitable, lower frequency. Figure 6.23 shows the calculated Bode plots for the system response. Note that, while the transient responses of the amplitude control system elements are the same as those of the phase control system, the steady state gain of these system elements i s not the same. For the phase control system the required steady state loop gain must be provided by the detector and compensator be-cause the gain of the RF chain and resonators i s unity. 162 6.5 Simulation Of The RF Feedback System 6.5.1 Description Of Method The IBM Continuous System Modeling Program (CSMP) ^ 3 was used to implement the simulation of the TRIUMF RF feed-back systems. This approach was taken because CSMP utilizes an application-oriented input language that permits a simu-lation to be prepared simply and directly from a block dia-gram representation of the system to be studied. An impor-tant feature of CSMP i s that i t automatically sorts the structure statements of a problem (on the user's demand) to establish the correct signal flow. To ensure that a l l inte-grator outputs are computed simultaneously at the end of an iteration cycle, centralized integration is used by CSMP. 1 IBM Application Program, System/360 Continuous System Modeling Program i360A-CXz16X1 Application Description. 1968. 2 IBM Application Program, System/360 Continuous System Modeling Program Users Manual, 1971. 3 IBM Application Program, System/360 Continuous System Modeling Program System Manual, 1967. 1 6 3 The structure of the simulation is simplified by using func-tion blocks (CSMP macros and Fortran subprograms) to model the major signal transfer functions. Table 6.8 l i s t s the function blocks used and Figures 6.24 and 6.30 show these blocks combined to model the EF feedback system. RES, RESA These two CSMP macros represent the transmission line-resonator transfer functions (see Equations 6.87 and 6.88). Because Ta»T2=Tb for both the fundamental and the third harmonic systems, these equations were i n i t i a l l y simplified to the form (RESA) Vx,y = AjAVgx,y(s) + A ?Albx,y(s) (6. 135) 1 + T a s Note that the last equation relates the vector components of the modulating signals. This form rather than that giving the amplitude and phase response was used because i t was desired to investigate not only the small signal behavior, but also the behavior for large s i g n a l s — f o r example, 'those occurring during beam turn-on. To convert from the vector representation to the polar (amplitude and phase) representation of the modulating sig-nals, the CSMP macro POLAR was used. The inverse transfor-mation was performed by the macro called CART. 1 6 4 RES transfer function for the transmission line-resonator system, exact EESA........transfer function for the transmission line-resonator system, approximate CART convert from polar to vector represen-tation POLAR........convert from vector to polar representation CBC Fortran subroutine subprogram for finding the Circulating Beam Current TRF1......... transfer function for the fundamental RF chain TRF3......... transfer function for the third harmonic RF chain FBA1,2,3,4 transfer functions for the feedback amplifiers in the four feedback loops VPEAK Fortran function subprogram for finding the peak RF voltage given VR1,PR 1,VR3,PR3 Table 6.8., CSMP Simulation Function Blocks CBC This i s the Fortran subroutine 1 which was used to cal-culate the instantaneous circulating beam current in the cyclotron. The method used to find this current i s based on the numerical method outlined in Section 6.3.1. To imple-ment this method, a history of the integrated resonator vol-1 A l i s t i n g of this Fortran subroutine i s given in Appendix "B". 165 tage i s stored in an array VINT (I,J). Each column of this array represents the integral at a different RF phase (1/2 of a fundamental RF cycle i s covered by the J columns). During each iteration the most recent values of the inte-grated voltage for each column are placed in the f i r s t row of the VINT, the elements of each row having f i r s t been shifted to next row. The elements of the I' th row are dis-carded. Given an iteration interval of dT seconds, VINT thus represents a history extending dT*(I-1) seconds into the past from the most recent iteration. VINT i s updated using the following second order formula VINT(I.J) = VINT(1,J) + ( 5 VMEM(l.J) 12 (6.136) + 2 VMEM(2,J) - 1 VMEM(3,J))*dT 3 2 where VMEfl i s an array in which the resonator voltage wave-forms for the most recent (1=1) and for the two preceeding int©rations are stored. The transit time i s found at each phase interval by searching the appropriate column of VINT in the manner shown in Figure 6.13. The instantaneous circulating current wave-form i s then found by u t i l i z i n g the value of the transit time at every phase and an array (BIP(J)) in which there i s stored a history of the i n i t i a l beam phase with respect to that of the RF reference. This last array i s required be-cause the beam i s injected over a range of phases referenced 166 to the peak of the RF fundamental voltage present on the resonators. Thus, even though the i n j e c t i o n phase—measured with respect to the peak of the fundamental RF resonator v o l t a g e — i s kept constant, the beam's phase with respect to the peak of the reference RF fundamental voltage, BIP(J), i s not necessarily constant. Because no variation i s allowed i n the phase width of the injected beam current pulses, the elements of BIP (J)- together with the t r a n s i t time at each RF phase are s u f f i c i e n t to determine the c i r c u l a t i n g current at that phase. That i s , the summation given by Equation 6.95 i s carried out to determine the instantaneous c i r c u l a t i n g current waveform as a function of the fundamental RF phase. This waveform i s then Fourier analysed to determine the i n -stantaneous amplitude and phase of the fundamental and t h i r d harmonic c i r c u l a t i n g currents. 6.5.2 Simulation Results 6.5.2.1 No Feedback This simulation was carried out to investigate the i n -teraction between the accelerating voltage parameters and the beam current i n the absense of feedback contr o l . Figure 6.24 gives a block diagram of the model used. For the pur-poses of t h i s study, the RF inputs to the resonators (Vg1,Pg1,Vg3,Pg3) were kept constant. Only the maximum energy resolution mode was investigated. 167 Fig. 6.24. Block Diagram For RF-beam Interaction 1 6 8 Figures 6.25(a) and 6.25(b) i l l u s t r a t e the ef f e c t of beam turn-on on the fundamental and third harmonic resonator voltages for two input beam pulse phase widths. As was to be expected from Section 6.3, the power coupled into the th i r d harmonic mode by the t h i r d harmonic component of the c i r c u l a t i n g beam current produces only a small increase i n the l e v e l of the t h i r d harmonic voltage. Figure 6.25(c) shows the steady state change in the fundamental and t h i r d harmonic voltages for increasing values of the peak input beam current (IBI). It i s interesting to note that, even without feedback, the resonator voltages are stable when a beam i s injected into the cyclotron. Thus a possible i n s t a b i l i t y — t h a t caused by a mutually i n t e r a c t i n g voltage and t h i r d harmonic beam current component—did not ex i s t . This i s due to the fac t that an increase in the t h i r d har-monic voltage produces a corresponding increase i n the fun-damental component of the c i r c u l a t i n g current. This loads, the fundamental mode s u f f i c i e n t l y to l i m i t the increase i n the t h i r d harmonic component of the c i r c u l a t i n g current and thus l i m i t s the increase i n the t h i r d harmonic voltage. It was also found that, for the maximum energy resolution mode, turning on the beam did not affect the phases of the funda-mental and t h i r d harmonic resonator voltages. In addition to simulating beam turn-on with the model shown i n Figure 6.24, t h i s model was also used to examine the ef f e c t of beam coupling between the fundamental and th i r d harmonic mode. This was accomplished by examining the 169 to AV r3 % (PW=45°) l AV r 3 % (PW=20°) P - A V % (PW-20°) AV x % (PW=45°) 0 . 0 2 5 . 0 TIME (MICROSEC.) ( a ) r e s o n a t o r v o l t a g e r e s p o n s e t o b e a m t u r n - o n o I b l (t>W=450)-v I b 3 (PW-45^—-2 • / V a / cn Q_ SE-CT: / / l b l (PW=20°) // 1 > CM CURREN1 0 I b ' 3 (PW=20°' CURREN1 0 a • a 0 . 0 2 5 . 0 5 0 . 0 TIME (MICROSEC.) (X10 f ( b ) v a r i a t i o n i n t h e m a g n i -t u d e o f t h e c i r c u l a t i n g b e a m c u r r e n t c o m p o n e n t s d u r i n a b e a m t u r n - o n 0 . 5 1 . 0 1 . 5 IBI (MILLIflMPS) ( c ) s t e a d y s t a t e c i r c u l a t i n g c u r r e n t c o m p o n e n t s a n d c h a n g e i n t h e f u n d a m e n t a l a n d t h i r d h a r m o n i c v o l t a g e a s a f u n c t i o n o f t h e p e a k i n j e c t i o n c u r r e n t ( I B ! ) F i g . 6 . 2 5 . B e a m t u r n - o n , t r a n s i e n t a n d s t e a d y s t a t e v a l u e s 170 effect of step changes in the RF input on the resonator vol-tages. Figure 6.26 shows the consequence of a step change in the amplitude of the fundamental resonator input voltage. From this Figure one can see that the change in the funda-mental resonator voltage amplitude (Vr1) i s of the same sign as the change in Vg1. The change in the third harmonic res-onator voltage i s , however, of the opposite sign. This occurs because an increase in the fundamental voltage de-creases the magnitude of both the fundamental and the third harmonic circulating currents. The resulting reduction in the power fed into the third harmonic resonator mode de-creases the third harmonic resonator voltage. The opposite effect occurs when the magnitude of Vg1 i s decreased. Figure 6.27 shows the effect on the fundamental and third harmonic voltages of a step change in the magnitude of the third harmonic driving voltage. Note that in this case the steady-state magnitude of the fundamental voltage i s decreased for both positive and negative going steps in the magnitude of Vg3. This occurs because, for the beam phase width used (45°), both these changes in Vg3 alter the RF resonator voltage waveform in a manner which increases the the fundamental component of the circulating current. Figures 6.26 and 6.27 also show that the circulating current reduces the overall resonator Q. Thus, from Figure 6.26, one can see that the time constant of the fundamental mode has been reduced from 50 microseconds to about 8 micro-seconds. From Figure 6.27 one can see that the time con-171 stant of the t h i r d harmonic mode has been reduced from about 15 microseconds to about 3 microseconds. The steady state c i r c u l a t i n g currents causing these changes i n the time con-stants may be seen in Figure 6.25(b). No mention has been made of the phase response to step changes in the RF driving voltages because the observed res-ponse (of the order of 10~ 6 degrees) was assumed a t t r i b u -table to innaccuracies in the simulation technigue. Figure 6.28 shows the response of the resonator phase to step changes i n the phase of the input voltage. As i s to be ex-pected, the time constants of the fundamental and t h i r d har-monic phase response were observed to be the same as those for the amplitude response. Since the RF waveform i s sym-metric for the maximum energy resolution mode, the resonator amplitude response i s the same for positive and negative step changes i n the phase of the input RF voltage. In addi-t i o n , the sign of the change in the resonator phase i s i n each case the same as the sign of the step change i n the input phase. 6.5.2.2 With Feedback The closed loop responses of the TRIUMF RF regulating systems were simulated using as feedback variables (a) the fundamental and t h i r d harmonic voltage (VR1,VR3) and the phase (PR1,PR3), and (b) the peak voltage (Vp), the voltage r a t i o (E), the fundamental phase (P1) and the phase of the th i r d 1 7 2 F i g . 6.26. Response To Step In Fundamental Voltage Source 173 P i g . 6.27. Response To Step In 3rd Harmonic Voltage 1 7 4 TIME (MICROSEC.) (XlO 1 ) F i g . 6.28. Response To Step In Input RF Phase harmonic with respect to the fundamental (P31). In the f i r s t case, the four feedback loops are coupled only by the RF-beam i n t e r a c t i o n . The response of t h i s system was found to be stable both with and without a beam present. Figure 6.29 shows the response of the resonator RF voltage parameters to step errors in a l l four feedback loops. Note the f i r s t overshoot i n the fundamental phase response; t h i s VOLTAGE REGULATION - Maximum energy resolution Mode of operation - No beam current - Feedback loops decoupled - 10% step in VR1 and VR3 at T=0 PHASE REGULATION - Maximum energy resolution Mode of operation - No beam current - Feedback loops decoupled - 1° step in PR1 and PR3 at T=0 0.0' 10.0 20.0 TIME (MICROSEC.) 30.0 Fig. 6.30. Schematic Of System With Detector Coupling 177 Maximum energy r e s o l u t i o n mode of operation +10% step i n VR1 at t=0 20.0 30.0 T (MICROSECJ F i g . 6.31. Detector Induced Coupling, Step In VE1 50.0 c h a r a c t e r i s t i c i s due to the capacitor coupling (low f r e -quency cut-off) used by t h i s feedback loop. Also, because of t h i s type of coupling, the steady state fundamental phase does not approach zero i f the input error consists of a step change. In the second case, the four EF feedback loops are coupled by both the beam and the measuring system. This 178 Maximum energy r e s o l u t i o n mode of operation 10% sten i n VU3 at t=0 20.0 30.0 (MICROSEC.) 50.0 F i g . 6.32. D e t e c t o r Induced C o u p l i n g , Step In VR3 system i s the one that i s to be used i n i t i a l l y at TRIUMF; F i g u r e 6.30 d e p i c t s the model used f o r i t ' s s i m u l a t i o n u s ing 1 A l i s t i n g o f t h i s CSMP s i m u l a t i o n program i s g i ven i n Appendix " B " . 179 CSMP. This study showed that t h i s system i s stable both with and without a c i r c u l a t i n g beam present. As shown by Figures 6.31 and 6.32, there i s considerable coupling be-tween the fundamental and t h i r d harmonic systems due to the measurement system. Decoupling these systems i n the manner outlined i n Section 6.1 would, therefore, aid considerably i n achieving the high s t a b i l i t y desired of t h i s system. 1 8 0 CHAPTER 7. RF PROBES The s t a b i l i t y and accuracy of the RF feedback control systems can be no better than the s t a b i l i t y and accuracy of the devices measuring the parameters to be controlled. Since i t i s desired to achieve a very high energy resolution of 2 parts i n 10 s, the design of suitable RF probes i s d i f -f i c u l t ; complicating the design of the probes are the f o l -lowing additional contraints which the probes must s a t i s f y ; (1) they must not degrade in the presence of radiation (The approximate flux at the center of the machine i s 3x10 7 neutrons/cm 2sec.) (2) they must not outgas to an extent which adversely a f f e c t s the vacuum i n the cyclotron ( 10 - 7 Torr) (3) they must not contain magnetic materials (4) they must be connected remotely. In addition to probes which function as the sensing elements for the feedback control systems, probes are also needed for diagnostic purposes, f o r example, to display the character-i s t i c s of the RF acceleration voltage and the current c i r c u -l a t i n g through the root of the resonator. These probes do not require the same high resolution required by those which are elements of the RF feedback systems. 181 7.1 Phase Probes The primary purpose of these probes i s to provide a s i g n a l from which to deduce the phase of the fundamental component of the resonator root current. This phase i s re-! i X Fig. 7.1. Resonator Section With Pick-Op Loop quired i n order to determine the r e l a t i o n s h i p between the resonator resonant frequency and the RF input frequency 1 2. A secondary purpose for these probes i s to monitor the res-onator c i r c u l a t i n g current (fundamental plus t h i r d harmonic) 1 M. Zach, Error Signal For Resonator Fine Tuning, TRI-DN-70-43, June 2,1970. 2 R. P o i r i e r , Determining the Resonator Frequency, TRIUMF, TRI-DN-73-8, Feb. 21,1973. 182 for supervisory purposes. In order to calculate the voltage induced i n a pick-up loop placed within the resonators i t i s f i r s t necessary to calculate the magnetic flux density (B) within the reson-ators. This may be done by noting that the current density in the one section of the resonator shown in Figure 7.1 i s J(x) = (Vo*L-i*Zo-i)cos (b*x) (amp./m.) (7.1) where Vo = peak resonator voltage = 100 kV Zo = c h a r a c t e r i s t i c impedance of one section = 3 8 ohms L- = width of one section = 1 m. b = .4817 rad./m. (fundamental mode) x = distance from the root (meters) Hence the current density at the root of the resonators (Jo) i s 262 amp./m.1 Since the resonators are excited i n a TEM mode, then jfl(x) | = |J(x) | (H i s the magnetic f i e l d i n t e n s i t y ) , and B (x) = uo*J (x) . (7. 2) Consequently, a loop having an area A (m2) placed at a point 1 A. Prochazka, Ph.D. Thesis, O.B.C., 1972. 183 x meters from the root w i l l have an induced voltage V = A*w*uo*Jo*cos (b*x) (7.3) where w .= EF frequency = 1.445x10* rad./sec. (fund.) uo = vacuum permeability = 1.257x10-6 henrys/m. A loop with an area of 1 cm2 w i l l therefore have 4.75 volts induced i f i t i s placed at the root of the resonators and the resonators are driven i n the fundamental mode. Pick-up loops 1 havinq an area of about 1 cm2 were placed at two positions in the resonators: at the root and at the current minimum of the t h i r d harmonic mode (1.09 meters from the root). The probe at the f i r s t l o c a tion i s for monitoring the current i n the resonator root; the probe at the second location provides the phase of the fundamental component of the c i r c u l a t i n g current. 7.2 Voltage Probes These are the probes which sample the RF waveform at the accelerating gap; In addition to providing signals for the RF control systems, these probes also provide a phase si g n a l for the chopper-buncher in the beam i n j e c t i o n l i n e , \ see T.RIUHF drawing D-978 Phase Probe Assembly and Deta i l s . 1 84 and a means whereby to monitor the RF waveform at various * points along the accelerating gap. The i n i t i a l design of the RF resonators envisaged a l l these functions being per-formed by a single type of RF probe. This probe, shown i n Figure 7.2, consists of a capacitive plate mounted i n the ground arm of each resonator, approximately 5 inches away from the hot arm t i p . D i r e c t l y above the probe i s a port through the vacuum tank to accommodate the e l e c t r i c a l and mechanical connections of the probe. The probe housing i n t h i s case has a dual function: f i r s t l y , to mount the capaci-tive pick-up plate plus i t s divider capacitors, and second-l y , as a means for mechanically adjusting the position of the resonator ground arm t i p s . It was found, however, that due to the mechanical i n s t a b i l i t y of the hot arms, t h i s type of probe was not adeguate for the measurement of the peak RF voltage. This may be seen by considering the equivalent c i r c u i t of Figure 7.3, from which one may write Vo = (Ct*Vr)/(Ct + Cp) = R*Vr (7.4) i where Ct = pick-up plate capacity = gaA/L Cp = step down capacity Vr = resonator voltage R = step down r a t i o = I O - 3 To f i r s t order, the f r a c t i o n a l change i n the output voltage may then be written 135 Ground Arm Adjustment Screw Vacuum-tight 13NC RF Si g n a l Feed-thru Vacuum Pore L i d F l e x i b l e S t a i n l e s s S t e e l Coupling Bellows Vacuum Port Reinforcing Ring Vacuum Tank L i d Bayonet Probe Connector Step-down Capacitors Ground Arm Surface Pick-up Capacitor Plate F i g . 7.2. Drawing Of Resonator Tip Voltage Probe 1 3 6 Vr RF Probe Peak Rectifier Fig. 7 . 3 . Voltage Probe—Equivalent Circuit dVo/Vo = (R*Cp/Ct) (dCt/Ct - dCp/Cp) + dVr/Vr (7.5) where the terms dCt/Ct and dCp/Cp produce a decalibration error. The most significant error in this case i s due to the variation of Ct caused by the vibration of the hot arm (L varying). That i s , since Ct=^>A/L, then dCt/Ct = -dL/L. (7.6) Since L=10cm., and dL can be as large as .05 cm., the conse-quent dec a l i b r a t i o n of the probe (.5%) greatly exceeds the required resolution of le s s than 1 part i n 10 s. However, t h i s probe i s s t i l l adequate for other uses: for example, the monitoring of the RF waveform at the accelerating gap. The measurement of the r a t i o (E) and the phase (?3) of the thi r d harmonic may also be carr i e d out by th i s probe. The reason for t h i s i s that a change i n the pick-up capacitance 187 (Ct) does not affe c t the magnitude r a t i o of, or the phase relationship between, the fundamental and thi r d harmonic components of the EF waveform. The most d i f f i c u l t problem in the design of these probes was that of providing f l e x i b i l i t y i n the mechanical adjustment of the ground arm t i p s while s a t i s f y i n g the con-s t r a i n t s outlined at the st a r t of t h i s chapter. I t was de-cided that a voltage probe was not required for every hot arm t i p ; thus only one probe per quadrant has been i n -s t a l l e d . At the remaining probe locations the pick-up plate and divider capacitors are replaced by an aluminum plug. To measure the peak of the EF voltage waveform, i t was decided to i n s t a l l four pairs of probes (one per quadrant) at the center of the cyclotron. Figure 7.4 i l l u s t r a t e s the layout of a pair of these probes 1. From t h i s drawing i t can be seen that, because the hot arms are connected at the i r t i p s , two advantages r e s u l t from placing the capacitive pick-up plates as shown; (1) due to the increased r i g i d i t y of the connected hot arms, the amplitude of the i r vibrations i s greatly re-duced (2) to f i r s t order, the decalibration of the probes due 1 TRIUMF Drawings D-654 and D-655 z TRIUMF Drawing E-310 183 Hoc Arm \ C o n n e c t i o n \ o2 Fig. 7.4. Center Region Voltage Probes to hot arm vibrations i s eliminated, i f the output of a pair of probes i s averaged. The averaging of the peak-rectified output of two probes i s accomplished by the c i r c u i t shown in Figure 7.5. The pro-cess of averaging the signals from two probes eliminates the decalibration due to the vibration of the hot arms, leaving as the major source of error the e f f e c t s due to changes i n the hot arm temperature. A:major source of thermal decalibration i s that due to the expansion and contraction of the center post. Consid-ering Figure 7.4, one may approximate the combined output by The f r a c t i o n a l change i n the output may then be written: Vo = [Vo1 + Vo2]/2 - (.5*Vr/Cp) (Ct2 + Ct2) . (7.7) dVo/Vo = ,5*(dCt1/Ct + dCt2/Ct) (7.8) 189 where i t was assumed that Ct1=Ct2=Ct. Since Ct= A/L, the decalibration becomes: dVo/Vo = -dL/L = s*Ta*dT/L (7.9) where dL = -ds Ta = ds/(s*dT) = c o e f f i c i e n t of expansion of the center post = 1.4x10-5/°F (aluminum) Consequently, since s= 4 1/2 i n . and L = 2 15/16 i n . , one has dVo/Vo = 2.14x10-s/°F. Another possible source of thermal decalibration errors i s the averaging c i r c u i t used (Figure 7.5). The action of t h i s c i r c u i t may be represented as Vo = f R P 2 1V1 + f R r i 1V2 (7.10) R n + R p 2 R 2 2 + R P I where R11 = Rgl+Rl R22 = Rg2+R2 Rp1 = R1 1 | | RI Rp2 = R22 || RI. Since R11=R22=R and Rl»R, i t i s then a very good approxim-ation to write: Vo = (V1 + V2)/2 (7.11) 190 Fig. 7.5- Signal Averaging C i r c u i t D i f f e r e n t i a t i n g the l a s t equation gives the change i n Vo due to small changes i n the values of the resistances: dVo/Vo = .5 (dV1+dV2)/V - .25(dRl1+dR22)/Rl + .5*R*dRl/R12 (7.12) The error may be written dVo/Vo = (,5*R/R1) (Trl-Tr)*dT (7.13) 1 9 1 where i t was assumed that the amplifier output impedances (Rg1,Rg2) could be neglected. Since i t i s possible to make Rl much larger than fi and also to obtain r e s i s t o r tempera-ture c o e f i c i e n t s (Trl,Tr) as low as several parts per mil-l i o n , one can see that the thermally induced d r i f t of the output of the averaging c i r c u i t may be neglected. The capacitive pick-up plate and the divider capacitors are two other sources of thermally induced errors. It. was found that the expansion of the capacitive pick-up plate could cause an error dVo/Vo = 2*Tc*dT (7.14) where Tc=.9x10~ 5/ oF (coef f i c i e n t of expansion of copper). The error for changes i n the temperature of the divider ca-pacitor was found to be dVo/Vo = Tcc*dT where Tec i s the temperature c o e f i c i e n t ot the capacitors used (about 10 ppm/°C). As the preceeding simple calculations have shown, i t i s extremely d i f f i c u l t , i f not impossible, to keep the long-term d r i f t of the voltage peak measuring c i r c u i t below the desired tolerance of less than 1 part in 10 s. To at t a i n t h i s degree of accuracy, i t i s f e l t that d i r e c t measurements w i l l have to be made on the cyclotron output beam. Note that beam measurements can be used to sense only r e l a t i v e l y slow changes i n the peak RF voltage. The reason for t h i s i s i that any change i n the parameters of the RF waveform w i l l be 192 averaged over a time span corresponding to the particle transit time through the machine (about 250 microseconds). Thus, for f u l l control of the peak RF voltage, low frequency signals from a beam probe must be combined with the high frequency signals from an RF voltage probe. The cut-off frequency for these probes must be chosen so that i t i s higher than the highest frequency component of the probe decalibration error. The effect of a high-pass f i l t e r on the output from the voltage probe may be seen by continuing the expansion of Eguation 7.4 to second order terms: dVo/Vo = dR/R + dVi/Vi + (dVi*dR)/Vo . (7.15) A suitable high-pass f i l t e r will remove the low frequency term dR/R, leaving dVo/Vo = (dVi/Vi) (1 + dR/R). (7.16) Thus one can see that the gain of a feedback system u t i l i -zing the filtered output of a voltage probe w i l l be modu-lated by the term dR/R. This effect may be ignored, how-ever, since |dR/R| i s much less than unity. 1 9 3 7.3 RF Measurements To test the amplitude regulation of the fundamental RF amplifier chain {the third harmonic amplifier was not yet completed at the time t h i s was written), the noise spectrum of t h i s system was measured. Three sets of measurements were made with the following conditions: (a) no feedback (b) feedback using drive modulation (c) feedback using screen modulation. These tests were made before the RF resonators were i n -s t a l l e d and therefore the transmission l i n e was terminated with the Brown Boveri dummy l o a d 1 . The RF signal was de-tected by a capacitive probe placed in the transmission l i n e near th i s load. The detector c i r c u i t used was the same as that used for the resonator probes; to simulate the e f f e c t of the resonators, the detector was followed by an RC c i r -c u i t having the same time constant as that measured f o r the resonator fundamental mode. These tests were conducted at an output power l e v e l of one megawatt and a l l signal l e v e l s were scaled to match those that would have appeared on the resonators. Figures 7.6 to 7.8 show the noise spectrum that was 1 This i s a 50 ohm, soda water dummy load and i s capable of absorbing 2 MW of cw power. 194 measured when no feedback was used to regulate the amplitude of the RF signal. From these plots i t i s obvious that at low frequencies a significant portion of the amplitude modu-lation of the RF signal occurs at frequencies that are mul-tiples of the 60 Hz, the power line frequency. At a modula-tion frequency of 60 Hz, the RMS amplitude of modulation i s 63.1 volts for a carrier level of 100 kV. Figures 7.9 to 7.11 show the noise spectra that were measured when two different feedback loops were used to reg-ulate the fundamental RF amplitude. The f i r s t method em-ployed modulation of the input drive to the RF amplifier chain. For measurements taken using this mode of regula-tion, a l l of the RF amplifier control settings were the same as those for the measurements taken without feedback pres-ent. As may be seen from these plots, drive modulation pro-duced a reduction of 56 dB in the RF noise level at a fre-quency of 60 Hz; this reduction corresponds closely to that expected from the open loop gain plotted in Figure 6.19. At this frequency the signal to noise amplitude ratio i s about 1 part in 106. In the second feedback mode the screen voltage of the i n i t i a l power amplifier (IPA) i s modulated. In this mode of operation the IPA i s driven to saturation (500 mA IPA grid current) to reduce the feedthrough of noise from the previ-ous stages in the RF amplifier chain. From Figure 7.9 one can see that at 60 Hz this technique produces a noise reduc-tion about 8 db (a factor of 2.51) larger than that obtained 195 (b) T3 63 90 . 100 120 140 , 160 180 2C( Frequency (Hz) 200-(a) A O 30 20 T3 10 25 30 Frequency (Hz) 50 . 606 * 63 30 dB =100 V on the resonators 1 MW output power 300 mA IPA grid current RMS noise amplitude Fig. 7.6. Noise Spectrum, Fundamental Mode, No Feedback (b) 30j (a) 40 _g4 c/s . . 2 5 - 3 0 . . 4 0 SO . 606o 200 — Frequency (Hz) — • 630 30 dB =IQO V on the resonators 1 MW output power 300 mA IPA grid current RMS noise amplitude Fig. 7.7. Noise Spectrum, Fundamental Mode, No Feedback 197 20 10 T3 --OL -a! 2k — — — — ' v W - - ' T -Frequency (Hz) 30 dB = 100 V on the resonators 1 MW output power 300 mA IPA grid current RMS noise amplitude oT3k Fig. 7 . 8 . Noise Spectrum, Fund Mode, No Feedback by drive modulation. Figure 7.10 shows that screen modula-tion yields a lower level of output noise up to a modulation frequency of about 600 Hz. Beyond this frequency drive mo-dulation gave better regulation. This i s due to the fact that the feedback amplifier used for this system was the same as the one previously used for the CRC (Center Region Cyclotron) screen modulating system. Because this feedback amplifier does not provide the proper feedback compensation, a large reduction in the noise level i s to be expected when a more suitable feedback amplifier i s installed. With the present system the signal to noise amplitude ratio at a mo-dulation frequency of 60 Hz i s 3.8 parts in 10 7. 198 (b) 40 30 20 10 63 "In* v •» Mr dill af- .on • f V Sere odu lat :io n \ \ \ V A A \ / V • i / \ V \ / \ \ i L \n "A h / /\ \ f w- H \ V J w V r 100 Frequency (Hz) 120 140 160 180 20i 200 (a) 13 30 20 10 0 / Dri ve Mc du la tion *> J Set eer i ^  od ul atloi I -W J / r / nJ / / 20 dB = 100 mV on the resonators 1 MW power output Screen Modulation with 500 mA IPA grid current Drive Modulation with 300 mA IPA grid current Fig. 7.9. Noise Spectrum, Fundamental, With Feedback (a) 40 30 200 — Frequency (Hz) > 630 20 dB = 100 mV on the resonators 1 MW power output Screen Modulation with 500 mA IPA grid current Drive Modulation with 300 mA IPA grid current Fig. 7.10. Noise Spectrum, Fundamental Mode, With Feedback 200 AO 30 « 20 "0 10 HSdrejeit Modulation Modulation 30 40 50 606: Frequency (Hz) 6.3k 20 dB = 100 mV on the resonators 1 MW output power Screen Modulation with 500 mA IPA grid current Drive Modulation with 300 mA IPA grid current Fig. 7.11. Noise Spectrum, Fund Mode, With F-B Examination of the drive modulation feedback error s i g n a l when the complete fundamental RF system (resonators included) was used showed that the average amplitude of the resonator voltage deviation was about 25 v o l t s . This obser-vation was made f o r an RF power l e v e l of about 1 MW and a resonator voltage of 100 kV. It was furthermore observed that by far the largest contribution to the voltage error was made by signals having frequencies in the k i l o h e r t z range. It i s f e l t that t h i s noise originates in the i n i t i a l t r a n s i s t o r i z e d stage of the fundamental RF amplifier chain because the noise l e v e l i s considerably reduced when screen 2 0 1 modulation i s used and the driving stages are driven to sa-turation. In f a c t , measurements have shown that the power l i n e related noise components contribute, at a maximum, about 5 volts (peak) to the resonator voltage error when drive modulation i s employed. Using screen modulation t h i s l e v e l has been reduced below the desired l e v e l of 1 v o l t . This was f a c i l i t a t e d by an extra 20 dB of loop gain obtained by adding a lag-lead network with an upper cut off frequency of about 500 Hz to the screen feedback amplifier. 2 02 CHAPTER 8. SUMMARY This thesis has been mainly concerned with the problems involved i n s t a b i l i z i n g the TRIUMF RF system to the degree required to achieve an energy resolution of 2 parts i n 10 s; a degree of s t a b i l i t y which has not been achieved by any other existing RF system having a comparable power output. The permissible l e v e l of modulation for any, single RF wave-form parameter has been presented. Because the actual de-vi a t i o n of the RF waveform from i t s desired value i s the resultant of deviations in a l l i t s parameters (these devia-tions are not s t a t i s t i c a l l y independant), these l e v e l s are the upper bounds on the parameters' noise spectra. In actual fact, therefore, the noise must be kept to much lower l e v e l s . Because the noise spectrum of the accelerated beam can be measured, i t was not considered useful to develope i n d e t a i l the s t a t i s t i c a l r e lationships between the noise spec-tra of the RF waveform parameters and that of the res u l t i n g ion beam. Due to the high degree of s t a b i l i t y required of the RF waveform, i t was recognized at an early date that the noise input to the TRIUMF RF system must be reduced to an absolute minimum for there to be any p o s s i b i l i t y of meeting the de-sire d c r i t e r i a . As one means to promote t h i s end, the use of resonator hot arm vibration dampers was proposed. Accor-ding to the then existing analysis of such vibration dam-pers, i t was expected that the vi b r a t i o n a l amplitude of the 203 hot arm panels would be reduced (at t h e i r mechanical reson-ant freguency) by a factor of about 46. Measurements showed that the actual reduction was a least an order of magnitude smaller. This discrepancy prompted the investigation of vibration dampers presented i n t h i s thesis. I t was found that when the response to a stochastic signal (band li m i t e d "White Noise") was considered, a reduction of only 5.8 could be expected for the same parameter values. This l a s t r e s u l t was obtained by a computer evaluation of the of the res-ponse; a useful approximation of the expected response was found that d i f f e r e d from the computer calculated values by l e s s than 2%. The s e n s i t i v i t y of a dynamic vibration damper to variations i n i t s parameters was calculated and i t was found that the parameter most d i f f i c u l t to reproduce a c c u r a t e l y — t h e c o e f f i c i e n t of viscous friction--was also the one that l e a s t affected the performance of the damper. The vibration dampers f i n a l l y constructed were found to reduce the amplitude of hot arm vibration by a factor of about 2, thereby reducing by a factor of 4 the possible power excursions of the RF amplifier. To further reduce the tuning error of the TRIUMF re-sonators, a pneumatic tuning system was devised. It was shown that the performance of t h i s system was c r i t i c a l l y affected by the length of the pneumatic l i n e s used. To minimize the length of these l i n e s , the electropneumatic transducers used i n t h i s system were placed approximately f i v e feet from the resonator tank surface. To s a t i s f y the 204 Modulation of the amplitude of the RF fundamental voltage (VI) (See Section 2.3) Modulating Frequency (kHz) ["fractional amplitude deviation] L Energy Resolution J =0 1.0 2. 2.0 6. 5.6 10.". 8.0 Modulation of the ratio of third harmonic voltage (E) (See Section 2.4) Modulating Frequency (kHz) =0. 2. 6. 10. Modulation of the third harmonic to fundamental phase (P3) (See Section 2.5) (a) ion phase = 0° Modulating Frequency (kHz) =0. 2. larger than 4. (b) ion phase = -2° Modulating Frequency (kHz) =0. 2. larger than 4. Modulation of the fundamental phase (PI) (See Section 2.6) For any ioh phase, the restrictions on the amplitude of the modulation of the fundamental phase disappear as the frequency of modulation approaches zero. fractional deviation in V3/V Energy Resolution 8.0 16. 45. 64. max. permitted amplitude 1.0° 1.3° 1.4° max. permitted amplitude .10° .15° .40° For frequencies larger than 2 kHz, ion phase 0°, the amplitude of modulation must be less than 6°. For frequencies larger than 2 kHz, ion phase -2°, the amplitude of modulation must be less than 4°. Table, 8.1. Summary—Permissible Levels of RF Modulation 205 transducer a i r requirements, i t was derermined that pneuma-t i c accumulators were required at their a i r supply inputs. A continuing problem with t h i s system has been the f a i l u r e of the tuning f o i l s actuated by t h i s system. To minimize the e f f e c t of such f a i l u r e s , the bellows driving the f o i l s are i n d i v i d u a l l y ported through the vacuum tank wall. Hence, should a par t i c u l a r f o i l or actuation bellows f a i l , i t can be isolated and then repaired when convenient. To aid i n the design and evaluation of the TRIUMF B F feedback regulating systems, the generalized modulation transfer function for a modulated c a r r i e r system was derived. I t i s shown that the amplitude, phase and frequen-cy modulation transfer functions, though not l i n e a r , can be close l y approximated by lin e a r functions for low l e v e l s of modulation. The functions thus derived are shown to corres-pond c l o s e l y with the observed system responses. It i s also shown that (for small l e v e l s of modulation) the coupling between amplitude and phase can safely be ignored. To evaluate the effect of beam coupling between the fundamental and the proposed t h i r d harmonic RF system, the combined system was d i g i t a l l y simulated. The res u l t s showed that, although beam coupling did e x i s t , t h i s e f f e c t was not inherently unstable for the beam currents considered (up to 10 times the maximum projected beam current). Because a regulating system cannot act to correct de-viations smaller than those the system i s capable of detec-ti n g , the problem of measuring the parameters of the TRIUMF 206 EF accelerating waveform i s a c r i t i c a l one. I t i s shown that the low frequency s t a b i l i t y of the RF voltage probes w i l l not meet the required standards of accurracy. Thus, to correct f o r low frequency errors i n the energy r e s o l u t i o n , d i r e c t measurements w i l l have to be made on the beam i t s e l f . Of course, these measurements w i l l only be useful for error frequencies having a period long with respect to the t r a n s i t time of a p a r t i c l e through the machine (about 2 50 microseconds). For higher modulating frequencies the probes are s u f f i c i e n t l y stable. In addition to the problem of s t a b i l i t y , the measure-ment system as presently constituted also creates a coupling problem. The d i f f i c u l t y arises because the parameters meas-ured: peak waveform voltage, t h i r d harmonic-to-fundamental voltage r a t i o , t h i r d harmonic-to-fundamental phase and fun-damental phase, are not the parameters that can be d i r e c t l y controlled. Instead, these are the fundamental and t h i r d harmonic voltage, and fundamental and t h i r d harmonic phase. If the parameters presently being detected are used as feed-back variables to control the l a t t e r , the system i s stable, but coupled. To improve the response the measured variables should be decoupled i n the the manner outlined i n t h i s the-s i s . To further improve the s t a b i l i t y of the TRIUMF RF sys-tem, a means must be provided to pinpoint the sources of RF system disturbances. As a f i r s t step toward t h i s end, a high resolution spectrum analyser (frequency range: under 3 207 Hz to over 50,000 Hz) should be used to determine the f r e -quencies at which the most s i g n i f i c a n t disturbances i n the EF waveform occur. Those disturbances occurring i n the v i -c i n i t y of 4 Hz and 10 Hz can be attributed to resonator v i -brations. In a s i m i l a r manner, those errors occuring at multiples of the powerline frequency may be detected. To associate these errors with s p e c i f i c devices, a c o r r e l a t i o n analyser should be employed. The sources of errors, once located, should be eliminated using the appropriate techni-que, that i s , by means of damping, i s o l a t i o n or l o c a l feed-back loops. 208 A SELECTED BIBLIOGRAPHY Atkinson, P. Feedback Control Theory for Engineers. Heineman Educational Books Ltd., 1968. Bewley, L. V. Two Dimensional Fields in Electrical Engineering. Dover, 1963. Burgerjon, J. J. and A. Strathdee, eds. AIP Conference Proceedings. American Institute of Physics, 1972. Collin, R. E. Field Theory of Guided laves. McGraw-Hill, 1960. Farago, P. S. Free Electron Physics. Penguin Books Ltd., 1970. Goodson, R. E. Viscous and Boundary Effects in Fluid Lines. Ph.D Thesis, Purdue, 1963. G r i f f i t h , B. W. Radio-Electronic Transmission Fundamentals. McGraw-Hill, 1962. Jameson, R. A. Analysis of a Proton Linear Accelerator RF System and Application to RF Phase Control. Los Alamos Scientific Laboratory, 1965. Kollath, P. Particle Accelerators. Pitman, 1967. Kolomansky, A. A. and A. N. Lebedev, Theory of Cyclic Accelerators. North-Holland Publishing Co., 1966. Ku, Y. H. Transient Circuit Analysis. Boston Technical Publishers, 1965. Livingood, J. J. Cyclic Partxcle Accelerators. Van Nostrand,1961. Martin, Thomas L., Jr. Electronic Circuits. Prentice-Hall, 1959. McCallum, P. A. and B. F. Brown. Laplace Transform Tables and Theorems. Holt, Rinehart and Hinston Minersky, N. Theory of Nonlinear Control Systems. McGraw-H i l l , 1969. Peskin, E. Transient and Steady-state Analysis of Electric Networks. Boston Technical Publishers, 1965. 2 0 9 P o r c e l l i , G. Transient Analysis of a Resonant Cavity, for a Separted Orbit Cyclotron. Chalk River, 1960. Prochazka, A. Design of the RF System for the TRIUMF Cyclotron. Ph.D Thesis, University of B r i t i s h Columbia, 1972. Ramo, S., J. R. Whimmery and T. VanDuzer. F i e l d s and Waves in Communication Electronics. John Wiley S Sons, 1967. Raven, F. H. Automatic Control Engineering. McGraw-Hill, 1968. Seely, Samuel. Electron-Tube C i r c u i t s . McGraw-Hill, 1958. S k i l l i n g , H. H. Electronic Transmission Lines. McGraw-H i l l , 1951. Stratton, J. A. Electromagnetic Theory. McGraw-Hill, 1941. Takahashi, Y., M. Rabins, and D. Auslander. Control and Dynamic Systems. Addison-Wesley, 1970. Terman, F. E. Electronic and Radio luaineering. McGraw-H i l l , 1955. Watkins, Bruce 0. Introduction to Control Systems. MacMillan, 1969. 210 Appendix "A" SUBROUTINE CBC(VB1,PR1R1,VR3,PB3R1,TIME,DT,TW,FC,EF,IBI 1PW,I1,P1,I3,P3) Q ********************************** ************************* C THIS PROG CALCULATES THE BEAM CURRENT AT TIME (T) C C C 11....MAG. OF FUND. CIRCULATING CURRENT (AMPS) C.....I3....MAG. OF 3RD. HARM. CIRCULATING CURRENT (AMPS) C . . . PL... PHASE OF FUND. COMP. OF CIRCULATING CURRENT C W.R.T. REF. (DEG.) C.....P3...•PHASE OF 3RD. HARM. COMP. OF CIRCULATING CURRENT C W.R.T. FUNDAMENTAL IN DEG. OF FUNDAMENTAL C C...EF....REQUIRED ENERGY GAIN (MEV) C.....VMEM....MATRIX WHICH STORES THE LAST VALUES OF VRES C OVER 1/2 CYCLE OF THE REF FUND. PERIOD C...VINT.... MATRIX WHICH STORES THE INTEGRATED VALUES OF TH C.....TP TRANSIT TIME FOR A GIVEN PHASE OF THE REF. VOLTAG C.....TIME....PRESENT TIME IN SIMULATION C.....FC ION ROTATION FREQUENCY (HERTZ) C BIP....VECTOR WHICH STORES THE INITIAL PHASE OF THE INP C BEAM CURRENT PULSE—PHASE IS W.R.T. THE REF FUND. WAVE C..... BP....INITIAL PHASE OF BEAM PULSE W.R.T. THE C PEAK OF THE FUNDAMENTAL VOLTAGE REFERENCE C.....M....NO. OF INTERVALS THE HALF-PERIOD OF THE FUND. IS C DIVIDED INTO C.....N....N*DT = LENGTH OF TIME FOR WHICH A HISTORY OF THE C INTEGRATED VOLTAGE IS KEPT C.....DT....TIME INCREMENT FOR CSMP INTEGRATION C.....TW....TIME AT WHICH ALL ARRAYS ARE WRITTEN ON UNIT 2 1TCC(100) C DVINT....INTEGRATED VOLTAGE REQUIRED FOR C A PARTICULAR PHASE C.....QN....NO. OF ELECTRONIC CHARGES ON ACCEL. PARTICLES C.....PS STARTING PHASE IN CALCULATIONS C VR1. . . . FUND. VOLTAGE AMP. (ON RESONATOR) C VR3....3RD HARM. VOLTAGE AMP. (ON RESONATOR) C......PR1R1....PHASE(DEG.) OF RESONATOR VOLTAGE W.R.T. REFER C......PR3R1 PHASE(DEG.) OF 3RD HARM. RES. VOLT. W.R.T. .. C W.R.T. REFERENCE (3RD HARM. DEG) C 1DEG = .01745329 RAD C 1 RAD. = 57.2957795 DEG. C......PW....PHASE WIDTH OF INPUT CURRENT PULSE C IN DEGREES OF FUNDAMENTAL C IBI....INPUT BEAM CURRENT (ASSUMED CONST. DURING PULSE C DP.... PHASE INTERVAL USED IN THESE CALCULATIONS C.....ICC.... NO. OF TIMES THE VALUE OF (WHERE) CHANGES C.....WHERE(K) .... LOGICAL VARIABLE WHICH IS TRUE AT C A PARTICULAR TIME IF CURRENT WAS BEING 211 C INJECTED AT THAT TIME C C C c********************************************************** C INITIALIZE VINT,VMEM,BIP,CURRENT,TP (AHRAYS) C********************************************************** REAL A,B,BIP (300),BP,C,D#DEL,DI,DP,DT,DVI,11,13 REAL DVINT,EF, EVI,FC,fl (181) ,IB (181) ,IBI,P,PI, PID2, REAL PR1R1,PR3R1,P8,P1,P3,QN,TF,TI,TIME,TM,T1,T2 REAL TP (2,9 1) ,V,VINT (300,91) , VMEM (3,91) ,VR1 ,VR3 REAL COSVC1 (181) ,SINVC1 (181) , COSVC3 (181) ,SINVC3 (181) INTEGER I,ICC,IDIF,IE,II,II1,112,J,K,M,N,MM,ITT LOGICAL AL, WHERE (305) TIM1=TIME+1. IF (TIM1.EQ.TIM2) GO TO 111 TIM2=TIM1 IF (TIME.GE. DT) GO TO 17 WRITE (6,999) TIME ,DT 999 FORMAT (1H ,»TIME = •,G12.5,T24,1DT = «,G12.5) TWL=TW-.1*DT TWH=TW+.1*DT C QN=1. READ (3) N,M READ (3) VINT READ (3) VMEM READ (3) BIP READ (3) TP READ (3) IB REWIND 3 NPD=M*2 NPDD=NPD+1 NN=N-1 NNN=N +1 MM=M+1 TM=NN*DT FCIBI=FC*IBI DPDEG=180./FLOAT(M) C r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C NOW CALCULATE THE REQUIRED CHANGE IN THE C INTEGRATED VOLTAGE (DVINT) FOR A PARTICLE IN TRANSIT Q************************************************************ c DVINT= (EF*1. E+6) / (4.*QN*FC) C C Q*********************************************************** C THIS IS THE END OF THE INITIALIZING SEGMENT C OF THE PEOGRAM C*********************************************************** C C INITIAL DATA IS STORED IN DATACBC1 2 1 2 C C PLACE MOST RECENT WAVEFORM IN THE FIRST ROW OF VMEM C FIRST SHIFTING THE VALUES IN VMEM DOWN ONE ROW C************************************************************ PI=3.1415926536 PID2=1.5707963268 C DEFINE THE PHASE INTERVAL DP=PI/FLOAT (H) DO 360 I=1,NPDD p=DP*PLOAT ( 1-1) COSVC1 (I) =COS (P) SINVC1 (I) = SIN (P) COSVC3 (I) =COS (3.*P) 360 SINVC3 (I) =SIN (3.*P) GO TO 111 17 DO 2 J=1,MM VMEM (3, J) =VMEM (2, J) 2 VMEM (2, J) =VMEM (1, J) PR1R1R=PR1R1*.01745329 PR3R1R=PR3R1*.01745329 DO 3 J=1,MH p=DP*FLOAT (J-1) 3 VMEM (1,J)=VR1*SIN(P + PR1R1R)+VR3*SIN(3.*P+PR3R1R) C Q*********************************************************** C C THE FOLLOWING IS THE PROGRAM SEGMENT FOR UPDATING C VINT(I,J) C*********************************************************** c DO 5 J=1,MM DVI=DT*(.4166666*VMEM(1,J)+.6666666*VMEM (2,J)-.0833333* IF(DVI) 220,221,221 220 DO 230 1=1,N 230 VINT(I,J)=0.0 GO TO 5 221 EVI=VINT (1,J)+DVI-DVINT IF(EVI) 240,240,241 240 DO 242 11=1,NN 242 VINT (NNH-II,J)=VINT(N-II,J) VINT (1, J) =VINT (2, J) +DV.I GO TO 5 241 DO 243 11=1,NN I=NNN-II VINT(I,J)=VINT(N-II,J) -EVI IF (VINT (I, J) .LT.O.) VINT (I, J) =0. 243 CONTINUE VINT (1, J)=DVINT 5 CONTINUE C*********************************************************** C SHIFT VALUES OF TP DOWN ONE ROW SO THAT C NEW VALUES OF TP MAI BE PLACED IN THE C FIRST ROW. ADD (DT) TO EACH VALUE IN THE SECOND C ROW SO THAT THE TIME IS MEASURED FROM THE 213 C PRESENT VALUE OF TIME C*********************************************************** DO 19 J=1,MM 19 TP (2,J) =TP (1,J) + DT C C THE FOLLOWING PROGRAM SEGMENT FINDS THE NEW TRANSIT C TIME (TP) C C*********************************************************** c TPM=NNN*DT DO 16 J=1,MM V=VINT (1,J) IF (V.EQ.DVINT) GO TO 301 IF(V.LT.DVINT.AND.V.GT.O.) GO TO 306 TP(1,J)=0. GO TO 16 301 I=IFIX (TP (2, J)/DT)+1 IF(I.GE.N) 1=11-1 IF (VINT (I, J) .LE. 0.0) GO TO 305 308 1=1+1 IF(I.LE.N.AND.VINT(I,J) .1E.0.0) GO TO 309 IF(I.EQ. N . AND. VINT (I, J).GT.0.0) GO TO 306 IF(I.LT.N.AND.VINT (I,J) .GT.O.0) GO TO 308 309 1=1-1 GO TO 307 305 IF(VINT (I,J)) 303,303,304 303 1=1-1 GO TO 305 304 IP (1-1) 310,310,307 307 DDT=V INT (I, J) *DT/ (VINT (1-1 ,J) - VINT (I,J) ) TP (1 ,J) =DT*FLOAT (1-1) +DDT GO TO 16 306 TP(1,J)=TPM GO TO 16 310 TP (1,J) =0.0 16 CONTINUE C c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C SHIFT THE VALUES IN BIP BACK ONE INTERVAL AND PLACE C THE MOST RECENT VALUE OF THE "BEAM INITIAL PHASE" INTO C FIRST LOCATION C c Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c DO 20 1=2,N 20 BIP (NNN + 1-1) =BIP (NNN-I) 5IP(1)=-PR1R1-BP+90. 122 CONTINUE C r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C NOW CALCULATE THE CIRCULATING CURRENT 214 C Q*********************************************************** C BIPHAX=BIP (1) BIPMIN=BIP (1) DO 500 1=2,N IF (BIP (I) .LT. BIPMIN) BIPMIN=BIP (I) IF (BIP (I) . GT. BIPMAX) BIPMAX=BIP(I) 500 CONTINUE ISTART= BIPMIN/DPDEG IF (ISTART.LT. 1) ISTART=1 ISTOP=(BIPMAX + PW)/DPDEG + 1 IF(ISTOP.GT.MM) ISTOP=MM DO 501 1=1,ISTART 501 IB (M+I)=0.0 DO 50 2 I=ISTOP,MM 502 IB(M + I)=0.0 TI=0.0 DO 102 I=ISTART,ISTOP P=DPDEG*FLOAT (1-1) IF(TP(1,I) .GT.TP(2,I) ) TP (1,1) =TP (2,1) IF(TM-TP(1,1)) 333,333,334 333 TP=TM-DT GO TO 3 35 334 TF=TP(1,I) 335 IF (TF.IE. 0.0) GO TO 101 II2=IFIX (TF/DT)+2 IF(II2.GT.N) II2=N IDIF=II2-1 IE=1 +IDIF DO 45 K=1,IE 45 MHERE (K)=BIP (K) . LE. P .AND. (BIP (K) +PW) .GE.P C rj * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C. WHERE (K) IS .TRUE. IF BIP (K) LIES IN THE C REGION WHERE CURRENT IS INJECTED C C CHECK WHETHER ALL OF WHERE (K) ARE TRUE OR C FALSE (CHECK FOR CHANGES IN THE VALUE OF WHERE) C. ... ,ICC=NO. OF TIMES THE VALUE OF (WHERE) CHANGES C*********************************************************** C ICC=0 DO 50 K=1,IDIF IF (WHERE (K) .AND. .NOT. WHERE (K+ 1) ) GO TO 492 IF (. NOT. WHERE (K) .AND. WHERE (K + 1) ) GO TO 492 GO TO 50 C Q*********************************************************** C CALCULATE THE CHANGE OVER POINTS (SOMEWHERE BETWEEN C BIP(J) AND BIP (J+1) C......TCC(20) IS THE ARRAY OF CHANGE-OVER TIMES C****************************************** ***************** c 215 492 ICC=ICC+1 C c*********************************************************** C CHECK WHETHER CHANGE-OVER IS DDE TO LEADING OR C TRAILING EDGE OF THE CURRENT PULSE C IT IS ASSUMED THE CURRENT PULSES OVERLAP AT SUCCESSIVE C TIME INTERVALS. Q*********************************************************** C IF (BIP (K) . LT. P. AND. BIP (K + 1) .LT. P) GO TO 491 TCC (ICC) = DT* (FLOAT (K-1) + ABS (BIP (K) -P) /ABS (BIP (K) -1BIP (K + 1 ) ) ) GO TO 50 491 TCC (ICC)=DT* (FLOAT (K-1) + ABS ( B X P ( K ) + PW-P) / ABS (BIP (K)-1BIP(K + 1) ) ) 50 CONTINUE IF (ICC. EQ. 0 . AND. WHERE ( 1) ) GO TO 340 IF(ICC.EQ.0.AND..NOT.WHERE(1) ) GO TO 341 GO TO 343 340 DI=FCIBI* (TF-TI) GO TO 100 341 DI=0. GO TO 100 C Q*********************************************************** C CHECK WHETHER TI PRECEDES OR FOLLOWS THE FIRST C CHANGE-OVER TIME C AL IS A LOGICAL VARIABLE TELLING WHETHER AT THE C STARTING TIME THE INTEGRAND IS POSTIVE OR ZERO C****** ***************************************************** 343 IF (TCC (1)-TI) 51,52,53 C C TI IS GREATER THAN TCC(1) C 51 TCC(1)=TI AL=.NOT.WHERE(1) GO TO 55 C C TI IS EQUAL TO TCC(1) C 52 TCC(1)=TI AL=.NOT.WHERE (1) GO TO 55 C C TI IS LESS THAN TCC(1) C 53 DO 54 IJ=1,ICC 54 TCC(ICC+2-IJ)=TCC(ICC+1-IJ) TCC (1)=TI ICC=ICC+1 AL=WHERE (1) 55 CONTINUE C r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *************** 216 C NOW ADD TF TO THE ARRAY TCC(20) C*********************************************************** c c IF (TCC (ICC) -TF) 60,68,64 C C TF IS GREATER THAN TCC (ICC) C 6 0 TCC(ICC + 1)=TF ICC=ICC + "1 GO TO 6 8 C C TCC (ICC) IS GREATER THAN TF C 64 TCC(ICC)=TF 68 CONTINUE C rj** **************************************** ***************** C WE NOW HAVE AN ARRAY OF TIMES TCC (I) AND A LOGICAL C VARIABLE (AL) WHICH TELLS WHETHER OR NOT THE VALUE OF T C INTEGRAND AT TCC(1) IS ZERO Q*********************************************************** C NOW FIND THE TIME INTERVAL (DEL) DURING WHICH C THE THE BEAM WAS BEING INJECTED C (DEL . LE. TF-TI ) C*********************************************************** c DEL=0. JFF=ICC-1 DO 70 J=1,JFF IF (AL) DEL=DEL + TCC (J+ 1) -TCC (J) AL=.NOT.AL 70 CONTINUE DI=DEL*FCIBI 100 IB (H + I) =DI GO TO 102 101 IB(M + I)=0.0 102 CONTINUE DO 345 J=1,M 345 IB (J) =-IB (M+J) IF(TIME.LT.T WL.OR.TIME.GT.TWH) GO TO 111 WRITE(4) N,M WRITE (4) VINT WRITE (4) VMEM WRITE (4) BIP WRITE(4) TP WRITE (4) IB 111 CONTINUE C c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C NOW FOURIER ANALYSE THE CIRCULATING CURRENT C FINDING THE MAGNITUDE AND PHASE OF THE C FUNDAMENTAL & 3RD HARMONIC COMPONENTS C** ********************************************* ************ 217 C DO 353 1=1,NPDD 353 H (I) =C0SVC1 (I) *IB (I) CALL QSF (DP, H, H, NPDD) AT=H (NPDD) /PI DO 351 1=1, NPDD 351 fl (I)=SINVC1 (I) *IB(I) CALL QSF (DP,H,H,NPDD) B1=H(NPDD)/PI DO 352 1=1,NPDD 352 H (I) =COSVC3 (I) *.IB (I) CALL QSF (DP, H, H, NPDD) A3=H(NPDD)/PI DO 354 1=1,NPDD 354 H (I) =SINVC3 (I) *IB (I) CALL QSF (DP, H, H, NPDD) B3=H (NPDD) /PI I1=SQ.RT (A1*A1 + B1*B1) IF(A1.EQ.0.0.AND.B1.EQ.0.0) B1=1. P1=ATAN2(A1,B1) P1=57.2957795*P1 I3=SQRT (A3*A3+B3*B3) IF(A3.EQ.0.0.AND.B3.EQ.0.0) B3=1. P3=ATAN2 (A3,B3) P3=57.2957795*P3 RETURN END 2 1 8 Appendix "B" CSMP P r o g r a m L i s t i n g - - a s a m p l e o f a t y p i c a l p r o g r a m SIMULATION OF THE RF-BEAM INTERACTION FUND. & 3RD HARMONIC F-B LOOPS USED RELATIVE 3RD HARMONIC AMPLITUDE AND PHASE USED AS F-B VARIABLES FOR CONVERTING TO CARTESIAN CO-ORD. TO POLAR CO-ORD. TITLE TITLE TITLE TITLE * *** MACRO # MACRO AX, AY=CART (AM, PA) *** PA IS IN DEGREES PARAD=PA*.0174532925 AX=AM*COS(PARAD) AY=AM*SIN(PARAD) ENDMAC * *** MACRO FOR CONVERTING * MACRO AM,PA=POLAR(AX,AY) *** PA IS IN DEGREES AM=SQRT (AX*AX+A Y* A Y) PROCEDURE PA=ANGLE(AX,AY) IF (AX) 2,1,2 1 PA=90. GO TO 3 2 X=AY/AX PARAD=ATAN (X) PA=PARAD*57.2 957795 13 3 CONTINUE ENDPRO ENDMAC * * *** MACRO FOR RESONATOR RESPONSE MACRO V=RES (V0,A1,A2,TA,TB,VG,IB) X=A1*VG+A2*IB Z=REALPL (V0,TA,X) V=REALPL(VO,TB,Z) ENDMAC * *** MACRO FOR THE RESONATOR RESPONSE * MACRO V=RESA(VO,A 1,A2,TA,VG,IB) X=A1*VG+A2*IB V=REALPL (V0,TA,X) ENDMAC * (APPROX) 2 1 9 *** MACHO REP. FUND. RF CHAIN * IC1 = INITIAL VALUE OF Y * IC2 = INITIAL VALUE OF THE DERIVATIVE OF Y *** USE—FIXED N MACRO Y=TRF1 (N, TL , E, WN , IC 1,IC2, X) Z=DELAY (N,TL,X) Y=CMPXPL (IC1,IC2,E,WN,Z) ENDMAC * * *** MACRO REP. FUND. RF CHAIN (APPROX.) * IC1 = INITIAL VALUE OF Y * IC2 = INITIAL VALUE OF THE DERIV. OF Y MACRO Y=TRF1A(E, W N,IC1, IC2, X) Y=CMPXPL(IC1,IC2,E,WN,X) ENDMAC * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE FUND. AMP. CONTROL SYSTEM. * MACRO Y = FBA1 (T1,T2,X) Y—LED'LAG (T1 , T2, X) ENDMAC * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE FUND. PHASE CONTROL SYSTEM. * MACRO Y=FBA2 (T1,T2,X) A=T2*T2 B=2.*T2 S2Z= (X-B*SZ-Z) /A SZ=INTGRL(0. ,S2Z) Z=INTGRL(0.,SZ) Y=T1*S2Z+SZ ENDMAC # * *** MACRO REPRESENTING THE 3RD HARMONIC DRIVER *** AMPLIFIER RESPONSE * MACRO Y=TRF3(T,X) Z=R EAL PL {0.,T, X) ¥=REALPL(0.,T,Z) ENDMAC * * *** MACRO REPRESENTING THE 3RD HARMONIC RF DRIVER (APPROX.) *** AMPLIFIER RESPONSE MACRO Y=TRF3A (A,X) Y=A*X ENDMAC * 220 * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE 3RD HARM PHASE CONTROL SYSTEM MACRO Y=FBA3 (T1,T2,T3,X) Z=LEDLAG(T1,T2,X) Y=REALPL(0.,T3,Z) ENDMAC * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE 3RD HARMONIC PHASE CONTROL SYSTEM (APPROX.) * MACRO Y=FBA3A (T1 ,T2,X) Y=LEDLAG (T1,T2,X) ENDMAC # * * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE 3RD HARMONIC AMP. CONTROL SYSTEM MACRO Y=FBA4 (T1,T2,T3,X) Z=LEDLAG(T1,T2,X) Y = REALPL(0.,T3,Z) ENDMAC * * *** MACRO REPRESENTING THE FB AMPLIFIER *** FOR THE 3RD HARMONIC AMPLITUDE CONTROL SYSTEM (APPROX.) * MACRO Y=FBA4A (T1,T2,X) Y=LED1AG (T1,T2,X) ENDMAC PARAMETER VR1REF=112500., VPREF=100000., ... EREF=.111111, P31REF=0., ... PS1REF=0., ... PR3REF=0., TFB1=50., TFB2=1600., ... TFB3=50., TFB4=50000., ... TFB5=10., TFB6=32000., TFB7=1., ... TFB8=10., TFB9=32000., TFB10=1., ... TL=1.3, E1=.6, WN1=1.38, ... T3"""• 5 f G 1=128. , ... G2=916., ... G,3=1.8E+4, ... G4=2.5E+3, ... A11=7.12, A12=-5230., TA1=50., ... A31=3.22, A32=-1540., ... TA3=15., TB3=2. PARAMETER S1=1., S2=0. * INITIAL * VB3BEF=EBEF*VB1BEF AG1SEF=VR1REF/A11 AG3BEF=VB3fiEF/A31 PG1REF=PR1REF VR1X0=VR1REF*COS(PR1REF) VR1Y0=VR1REF*SIN(PR1REF) VR3X0=VR3REF*COS (PR3REF) VR3Y0=VR3R£F*SIN (P13REF) * * DYNAMIC * *** ERROR DETECTORS * VP=VPEAK(VR1,PR1,VR3,PR3) EVP=VPREF-VP EP1=PR1REF-PR1 P31=PR3-3.*PR1 EP31=P31REF-P31 E=VR3/VR1 EE=EREF-E * *** F-B GAIN * EVPP=G1*EVP EP11=G2*EP1 EP33=G3*EP31 EEE=G4*EE * *** FEEDBACK AMPLIFIERS * V TFB=FBA1(TFB1,TFB2,EVPP) P1FB=FBA2(TFB3,TFB4,EP11) P3FB=FBA3{TFB5,TFB6,TFB7,EP33) V3FB=FBA4(TFB8,TFB9,TFB10,EEE) * *** RF AMPLIFIERS * V1F BB=T RF1 {10,TL,E1,HN1,0.,0.,V1FB) P1FBB=TRF1 (10,TL,E1,WN1,0.,0.,P1FB) P3FBB=TRF3(T3,P3FB) V3FBB=TRF3(T3, V3FB) * *** ADD RESONATOR INPUT AMP £ PHASE * AG1=AG1REF+V1FBB PG1=PB1BEF+P1FBB PG3=PR3REF+P3FBB AG3=AG3REF+V3FBB *** CONVERT TO CARTESIAN CO-ORD * V1X,V1Y=CART(AG1,PG1) 222 NOTE: EFFECT I B l x A IB1Y IB3X IB3Y Substitute 0. 'TBI,PB1,IB3,PB3=CBC(VR1I,PRl1,VR33,PR33, TIME,DELT,TW,FC,EF,IBI,BP,PW) IB1X,IB1Y=CART(IB1,PB1) J.B3X,IB3Y=CART(IB3,PB3) V3X,V3Y=CART <AG3,PG3). *** RESONATOR RESPONSE * VR1X=RESA(VR1X0,A11,A12,TA1,V1X,0.) < VR1Y=RESA(VR1Y0,A11,A12,TA1,V1Y,0.) < VR3 X= RES (VR3X0 , A31 , A3 2 ,T A3 , T B3 , V3X , 0 . VR3Y=RES (VR3YG,A31,A32 ,TA3,TB3, V3Y, 0. J " 3 -* *** CONVERT TO POLAR CO-ORD VR 11,PR 1 1 = POLA R (VR1X,VR1Y) VR33,PR33=POLAR (VR3X ,VR3Y) * 4 *** ERROR INTRODUCTION * X=STEP (. 1) XR=.1*X VR1=VR11+S1*XR*VR1REF PR1=PR11+S1*XR*10. VR3=VR33+S2*XR*VR3REF PR3=PR33+S2*XR*10. * *** OUTPUT VARIABLES * VR3DEV= (VR3-VR3REF)/VR3REF VR1DEV=(VR1-VR1REF) /VR TR EF PR1DEV=PR1-PR1REF PR3DEV=PR3-PR3REF * *** TERMINATING SECTION * TIMER DELT=.01, FINTIM=.3, OUTDEL=.01, PRDE.L=.01 PRINT VR1DEV,PR1DEV,VR3DEV,PR3DEV,AG1,PG1,AG3,PG3, .., EVP,EP1,EP31,EE PETPLOT VR1,PR1,VR3,PR3,VP,P31,E * METHOD ADAMS * END PARAMETER S1=0., S2=1. END STOP ENDJOB TO INCLUDE THE OF THE BEAM, ADD: for 

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