Accelerator Physics Particle Acceleration. G. A. Krafft Old Dominion University Jefferson Lab Lecture 10

Transcription

1 Accelerator Physics Particle Acceleration G. A. Krafft Old Dominion University Jefferson Lab Lecture 1 Graduate Accelerator Physics Fall 17

2 RF Acceleration Characteriing Superconducting RF (SRF) Accelerating Structures Terminology Energy Gain, R/Q, Q, Q L and Q ext RF Equations and Control Coupling Ports Beam Loading RF Focusing Betatron Damping and Anti-damping Graduate Accelerator Physics Fall 17

3 Terminology 1 CEBAF Cavity 5 Cell Cavity Cells 1 DESY Cavity 9 Cell Cavity Graduate Accelerator Physics Fall 17

4 Modern Jefferson Lab Cavities (1.497 GH) are optimied around a 7 cell design Typical cell longitudinal dimension: λ RF / Phase shift between cells: π Cavities usually have, in addition to the resonant structure in picture: (1) At least 1 input coupler to feed RF into the structure () Non-fundamental high order mode (HOM) damping (3) Small output coupler for RF feedback control Graduate Accelerator Physics Fall 17

5 Graduate Accelerator Physics Fall 17

6 Some Fundamental Cavity Parameters Energy Gain d mc ee x t, t v dt For standing wave RF fields and velocity of light particles, cos,, cos / E x t E x t mc e E d RF RF i ee / RF e c.c. = V ee / Normalie by the cavity length L for gradient Vc Eacc MV/m L c RF Graduate Accelerator Physics Fall 17

7 Shunt Impedance R/Q Ratio between the square of the maximum voltage delivered by a cavity and the product of ω RF and the energy stored in a cavity (using accelerator definition) R Vc Q RF stored energy Depends only on the cavity geometry, independent of frequency when uniformly scale structure in 3D Piel s rule: R/Q ~1 Ω/cell CEBAF 5 Cell CEBAF 7 Cell DESY 9 Cell 48 Ω 76 Ω 151 Ω Graduate Accelerator Physics Fall 17

8 Unloaded Quality Factor As is usual in damped harmonic motion define a quality factor by Q energy stored in oscillation energy dissipated in 1 cycle Unloaded Quality Factor Q of a cavity Q RF stored energy heating power in walls Quantifies heat flow directly into cavity walls from AC resistance of superconductor, and wall heating from other sources. Graduate Accelerator Physics Fall 17

9 Loaded Quality Factor When add the input coupling port, must account for the energy loss through the port on the oscillation 1 1 total power lost 1 1 Qtot QL RF stored energy Qext Q Coupling Factor Q Q 1 for present day SRF cavities, QL Q 1 ext It s the loaded quality factor that gives the effective resonance width that the RF system, and its controls, seen from the superconducting cavity Chosen to minimie operating RF power: current matching (CEBAF, FEL), rf control performance and microphonics (SNS, ERLs) Graduate Accelerator Physics Fall 17

10 Q vs. Gradient for Several 13 MH Cavities 1 11 Q 1 1 AC7 AC7 AC73 AC78 AC76 AC71 AC81 Z83 Z Courtesey: Lut Lilje E acc [MV/m] Graduate Accelerator Physics Fall 17

11 E acc [MV/m] E acc vs. time BCP EP 1 per. Mov. Avg. (BCP) 1 per. Mov. Avg. (EP) Courtesey: Lut Lilje Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan- Jan-1 Jan- Jan-3 Jan-4 Jan-5 Jan-6 Graduate Accelerator Physics Fall 17

12 RF Cavity Equations Introduction Cavity Fundamental Parameters RF Cavity as a Parallel LCR Circuit Coupling of Cavity to an rf Generator Equivalent Circuit for a Cavity with Beam Loading On Crest and on Resonance Operation Off Crest and off Resonance Operation Optimum Tuning Optimum Coupling RF cavity with Beam and Microphonics Q ext Optimiation under Beam Loading and Microphonics RF Modeling Conclusions Graduate Accelerator Physics Fall 17

13 Introduction Goal: Ability to predict rf cavity s steady-state response and develop a differential equation for the transient response We will construct an equivalent circuit and analye it We will write the quantities that characterie an rf cavity and relate them to the circuit parameters, for a) a cavity b) a cavity coupled to an rf generator c) a cavity with beam d) include microphonics Graduate Accelerator Physics Fall 17

14 RF Cavity Fundamental Quantities Quality Factor Q : Q W P diss Energy stored in cavity Energy dissipated in cavity walls per radian Shunt impedance R a (accelerator definition); R a V P c diss Note: Voltages and currents will be represented as complex quantities, denoted by a tilde. For example: it i V t Re V t e V t V e c c c c where Vc Vc is the magnitude of V c and is a slowly varying phase. t Graduate Accelerator Physics Fall 17

15 Equivalent Circuit for an RF Cavity Simple LC circuit representing an accelerating resonator. Metamorphosis of the LC circuit into an accelerating cavity cell. Chain of weakly coupled pillbox cavities representing an accelerating cavity. Chain of coupled pendula as its mechanical analogue. Graduate Accelerator Physics Fall 17

16 Equivalent Circuit for an RF Cavity An rf cavity can be represented by a parallel LCR circuit: i t V c() t V ce Impedance Z of the equivalent circuit: Z 1 1 ic R il 1 Resonant frequency of the circuit: 1/ LC Stored energy W: Vc W C Vc L il L L Graduate Accelerator Physics Fall 17

17 Equivalent Circuit for an RF Cavity Average power dissipated in resistor R: From definition of shunt impedance Quality factor of resonator: R a Vc Ra P diss Q P diss Vc R W CR Pdiss R Note: Z R 1 iq 1 For Z R 1iQ 1 Wiedemann Graduate Accelerator Physics Fall 17

18 Cavity with External Coupling Consider a cavity connected to an rf source A coaxial cable carries power from an rf source to the cavity The strength of the input coupler is adjusted by changing the penetration of the center conductor There is a fixed output coupler, the transmitted power probe, which picks up power transmitted through the cavity Graduate Accelerator Physics Fall 17

19 Cavity with External Coupling Consider the rf cavity after the rf is turned off. Stored energy W satisfies the equation: P P P P Total power being lost, P tot, is: tot diss e t P e is the power leaking back out the input coupler. P t is the power coming out the transmitted power coupler. Typically P t is very small P tot P diss + P e Recall W W Q dw dt P diss Energy in the cavity decays exponentially with time constant: Q / Q L dw dt P tot t W QL W We QL L L / Ptot Graduate Accelerator Physics Fall 17

20 Decay rate equation suggests that we can assign a quality factor to each loss mechanism, such that where, by definition, P P P W W tot diss e Q Q Q L Q e W P e Typical values for CEBAF 7-cell cavities: Q =1x1 1, Q e Q L =x1 7. e Graduate Accelerator Physics Fall 17

21 Have defined coupling parameter : e P P diss Q Q e and therefore 1 (1 ) Q QL Wiedemann 16.9 It tells us how strongly the couplers interact with the cavity. Large implies that the power leaking out of the coupler is large compared to the power dissipated in the cavity walls. Graduate Accelerator Physics Fall 17

22 Cavity Coupled to an RF Source The system we want to model. A generator producing power P g transmits power to cavity through transmission line with characteristic impedance Z Z Z Between the rf generator and the cavity is an isolator a circulator connected to a load. Circulator ensures that signals reflected from the cavity are terminated in a matched load. Graduate Accelerator Physics Fall 17

23 Transmission Lines In L Vn1 L Vn L VN I N In1 I n C C C Z Inductor Impedance and Current Conservation V V ili n1 n n1 I I icv n1 n n Graduate Accelerator Physics Fall 17

24 Transmission Line Equations Standard Difference Equation with Solution V V e, I I e i 1 i i 1 Continuous Limit ( N ) n in n sin / in V e i LI e I e icv LC V L / CI e V L / CI e i/ i/ LC k LC v 1/ LC V x, t V e L/ CI e Z I e it kx it kx it kx V x, t V e L/ CI e Z I e it kx it kx it kx Graduate Accelerator Physics Fall 17

25 Cavity Coupled to an RF Source Equivalent Circuit V, I Z ik I I V, I 1:k RF Generator + Circulator Coupler Cavity Coupling is represented by an ideal (lossless) transformer of turns ratio 1:k Graduate Accelerator Physics Fall 17

26 Cavity Coupled to an RF Source From transmission line equations, forward power from RF source Reflected power to circulator V V /Z /Z Transformer relations V kv k V V c k Considering ero forward power case and definition of β i i / k I I / k I / k V / k Z c k c V / Z / V / / c R R k Z Graduate Accelerator Physics Fall 17

27 Cavity Coupled to an RF Source Loaded cavity looks like i () t g Re I g k t i t ie Wiedemann Fig R R R Z g Wiedemann Zg k Z 16.1 Effective and loaded resistance Ra RL Reff R R Z R 1 eff Solving transmission line equations g 1Vc 1Vc V kzic V kzic k k Graduate Accelerator Physics Fall 17

28 Forward Power Reflected Power Powers Calculated 1 1 c V c L Ra V Pg iq Q 8Z k 4 1 c V c 1 1 L Ra Power delivered to cavity is as it must by energy conservation! 1 1 V Prefl iq Q 8Z k P P P Vc Vc g refl diss Ra Ra Graduate Accelerator Physics Fall 17

29 Graduate Accelerator Physics Fall 17 Some Useful Expressions Total energy W, in terms of cavity parameters Total impedance (1 ) 4 1 (1 ) 1 diss g diss L g g L Q P W P P Q Q W P Q Q W P Q (1 ) TOT g a TOT Z Z Z R Z iq

30 When Cavity is Detuned Define Tuning angle : Note that: tan QL QL for 4 Q 1 W= (1+ ) 1+tan P g Wiedemann 16.1 P diss 4 1 (1 ) 1 tan P g Graduate Accelerator Physics Fall 17

31 Optimal β Without Beam. Optimal coupling: W/P g maximum or P diss = P g which implies for Δω =, β = 1 This is the case called critical coupling. Reflected power is consistent with energy conservation: P P P P refl g diss refl P. On resonance: g (1 ) 1 tan W P P diss refl 4 (1 ) 4 (1 ) 1 1 Q P P g P g g Dissipated and Reflected Power Graduate Accelerator Physics Fall 17

32 Equivalent Circuit: Cavity with Beam Beam through the RF cavity is represented by a current generator that interacts with the total impedance (including circulator). Equivalent circuit: dv V di ic C, ir, Vc L dt R / dt c c L i g the current induced by generator, i b beam current Differential equation that describes the dynamics of the system: d Vc dvc RL d Vc ig ib dt Q dt Q dt L L L Graduate Accelerator Physics Fall 17

33 Kirchoff s law: Cavity with Beam il ir ic ig ib Total current is a superposition of generator current and beam current and beam current opposes the generator current. Assume that voltages and currents are represented by complex phasors g b c Re c it Re g it Re b it V t V e i t i e i t i e where is the generator angular frequency and are complex quantities. V, i, i c g b Graduate Accelerator Physics Fall 17

34 Voltage for a Cavity with Beam Steady state solution RL (1 i tan ) Vc ( ig ib ) where is the tuning angle. Generator current P P P ig I 4 k Z R R g g g a For short bunches: beam current. i I b where I is the average Wiedemann Graduate Accelerator Physics Fall 17

35 Voltage for a Cavity with Beam At steady-state: RL / RL / V i i (1 itan ) (1 itan ) c g b RL i RL or Vc ig cos e ib cos e i or V V cos e V cos e c gr br i i or V V V c g b V V voltages on resonance and gr br RL i RL V V g b g i b are the generator and beam-loading are the generator and beam-loading voltages. Graduate Accelerator Physics Fall 17

36 Voltage for a Cavity with Beam Note that: Vgr Pg RL Pg RL for large 1 V R I br L Wiedemann Wiedemann 16. Graduate Accelerator Physics Fall 17

37 Voltage for a Cavity with Beam V V cos e g V V cos e b gr br i i Im( V g ) V gr cos i e V gr Re( V g ) As increases, the magnitudes of both V g and V b decrease while their phases rotate by. Graduate Accelerator Physics Fall 17

38 Example of a Phasor Diagram V br V b V g I acc b I b V c I dec Wiedemann Fig Graduate Accelerator Physics Fall 17

39 On Crest/On Resonance Operation Typically linacs operate on resonance and on crest in order to receive maximum acceleration. On crest and on resonance V br I b V c V gr V V V c gr br where V c is the accelerating voltage. Graduate Accelerator Physics Fall 17

40 More Useful Equations We derive expressions for W, V a, P diss, P refl in terms of and the loading parameter K, defined by: K=I R a /( P g ) From: V gr g L V R I br 1 L V V V c gr br P R V c g L W P diss P R 4 (1 ) a a diss 1 1 K 1 c K 1 g Q I V I R P 4 (1 ) IV P 1 1 K K K P g ( 1) K P P P I V P P ( 1) P refl g diss a refl g g Graduate Accelerator Physics Fall 17

41 More Useful Equations For large, 1 P ( V I R ) g c L 4RL 1 P ( V I R ) refl c L 4RL For P refl = (condition for matching) R L and P V I M c M I V V I M M c c g M M 4 Vc I Graduate Accelerator Physics Fall 17

42 Example For V c =14 MV, L=.7 m, Q L =x1 7, Q =1x1 1 : Power I = I = 1 A I = 1 ma P g 3.65 kw 4.38 kw kw P diss 9 W 9 W 9 W I V c W 1.4 kw 14 kw P refl 3.6 kw.951 kw ~ 4.4 W Graduate Accelerator Physics Fall 17

43 Off Crest/Off Resonance Operation Typically electron storage rings operate off crest in order to ensure stability against phase oscillations. As a consequence, the rf cavities must be detuned off resonance in order to minimie the reflected power and the required generator power. Longitudinal gymnastics may also impose off crest operation in recirculating linacs. We write the beam current and the cavity voltage as I V b I e V e i b and set i c c c c The generator power can then be expressed as: P Vc (1 ) IR L IR L g 1 cosb tan sinb RL 4 Vc Vc Wiedemann Graduate Accelerator Physics Fall 17

44 Off Crest/Off Resonance Operation Condition for optimum tuning: tan IR V Condition for optimum coupling: Minimum generator power: c L IR sin a opt 1 cos V c b b P g,min V c R a opt Wiedemann Graduate Accelerator Physics Fall 17

45 Clarifications from Last Time On Crest, RI 1 RI a 1 1 Pg RL Pg L Vc Pg RL RLI Pg RL 1 K Off Crest with Detuning P P g g Z I Z k i = 8 V i i I i 1 tan cos sin c g b b RL Zk Vc IR L IR L g 1 cos tan sin b b RL Vc Vc Graduate Accelerator Physics Fall 17

46 Off Crest/Off Resonance Power Typically electron storage rings operate off crest in order to ensure stability against phase oscillations. As a consequence, the rf cavities must be detuned off resonance in order to minimie the reflected power and the required generator power. Longitudinal gymnastics may also impose off crest operation in recirculating linacs. We write the beam current and the cavity voltage as I V b I e V e i b and set i c c c c The generator power can then be expressed as: P Vc (1 ) IR L IR L g 1 cosb tan sinb RL 4 Vc Vc Wiedemann Graduate Accelerator Physics Fall 17

47 Optimal Detuning and Coupling Condition for optimum tuning with beam: tan IR V Condition for optimum coupling with beam: Minimum generator power: c L IR sin a opt 1 cos V c b b P g,min V c R a opt Wiedemann Graduate Accelerator Physics Fall 17

48 C75 Power Estimates G. A. Krafft Graduate Accelerator Physics Fall 17

49 P (kw) 1 GeV Project Specs 7-cell, 15 MH, 93 ohms kw MV/m, 46 ua, 5 H, deg 1. MV/m, 46 ua, H, deg 1. MV/m, 46 ua, 15 H, deg 1. MV/m, 46 ua, 1 H, deg 1. MV/m, 46 ua, H, deg Qext (1 6 ) Delayen and Krafft TN mua*15 MV=6.8 kw Graduate Accelerator Physics Fall 17

50 P (kw) 7-cell, 15 MH, 93 ohms MV/m, 46 ua, 5 H, deg 4. MV/m, 46 ua, 5 H, deg 3. MV/m, 46 ua, 5 H, deg. MV/m, 46 ua, 5 H, deg 1. MV/m, 46 ua, 5 H, deg Qext (1 6 ) Graduate Accelerator Physics Fall 17

51 Assumptions Low Loss R/Q = 93*5/7 = 645 Ω Max Current to be accelerated 46 µa Compute and 5 H detuning power curves 75 MV/cryomodule (18.75 MV/m) Therefore matched power is 4.3 kw (Scale increase 7.4 kw tube spec) Q ext adjustable to (if not need more RF power!) Graduate Accelerator Physics Fall 17

52 14 Power vs. Qext 1 5 H detuning P g (kw) detuning Q ext ( 1 6 ) Graduate Accelerator Physics Fall 17

53 Frequency (H) Probability Density RF Cavity with Beam and Microphonics f f f m The detuning is now: tan Q tan Q L L f f where f and f m is the static detuning (controllable) is the random dynamic detuning (uncontrollable) Time (sec) Probability Density Medium CM Prototype, Cavity #, 6MV/m 4 samples Peak Frequency Deviation (V) Graduate Accelerator Physics Fall 17

54 Q ext Optimiation with Microphonics P Vc (1 ) ItotR L ItotR L 1 cos tan sin RL 4 Vc V c g tot tot f tan Q L f where f is the total amount of cavity detuning in H, including static detuning and microphonics. Optimiing the generator power with respect to coupling gives: f opt ItotRa where b cos tot V ( b 1) Q b tan f c where I tot is the magnitude of the resultant beam current vector in the cavity and tot is the phase of the resultant beam vector with respect to the cavity voltage. tot Graduate Accelerator Physics Fall 17

55 Graduate Accelerator Physics Fall 17 Correct Static Detuning To minimie generator power with respect to tuning: The resulting power is tan tot f f b Q 1 (1 ) 4 c m g a V f P b Q R f c m g opt opt a opt c opt a V f P b b Q R f V b R

56 Optimal Q ext and Power Condition for optimum coupling: and m ( b1) Q f In the absence of beam (b=): opt f opt V c f m Pg b 1 ( b 1) Q R a f and P opt f m 1Q f V f opt c m g 1 1 Q R a f Graduate Accelerator Physics Fall 17

57 Problem for the Reader Assuming no microphonics, plot opt and P g opt as function of b (beam loading), b=-5 to 5, and explain the results. How do the results change if microphonics is present? Graduate Accelerator Physics Fall 17

58 Example ERL Injector and Linac: f m =5 H, Q =1x1 1, f =13 MH, I =1 ma, V c = MV/m, L=1.4 m, R a /Q =136 ohms per cavity ERL linac: Resultant beam current, I tot = ma (energy recovery) and opt =385 Q L =.6x1 7 P g = 4 kw per cavity. ERL Injector: I =1 ma and opt = 5x1 4! Q L = x1 5 P g =.8 MW per cavity! Note: I V a =.8 MW optimiation is entirely dominated by beam loading. Graduate Accelerator Physics Fall 17

59 RF System Modeling To include amplitude and phase feedback, nonlinear effects from the klystron and be able to analye transient response of the system, response to large parameter variations or beam current fluctuations we developed a model of the cavity and low level controls using SIMULINK, a MATLAB-based program for simulating dynamic systems. Model describes the beam-cavity interaction, includes a realistic representation of low level controls, klystron characteristics, microphonic noise, Lorent force detuning and coupling and excitation of mechanical resonances Graduate Accelerator Physics Fall 17

60 RF System Model Graduate Accelerator Physics Fall 17

61 RF Modeling: Simulations vs. Experimental Data Measured and simulated cavity voltage and amplified gradient error signal (GASK) in one of CEBAF s cavities, when a 65 A, 1 sec beam pulse enters the cavity. Graduate Accelerator Physics Fall 17

62 Conclusions We derived a differential equation that describes to a very good approximation the rf cavity and its interaction with beam. We derived useful relations among cavity s parameters and used phasor diagrams to analye steady-state situations. We presented formula for the optimiation of Q ext under beam loading and microphonics. We showed an example of a Simulink model of the rf control system which can be useful when nonlinearities can not be ignored. Graduate Accelerator Physics Fall 17

63 RF Focussing In any RF cavity that accelerates longitudinally, because of Maxwell Equations there must be additional transverse electromagnetic fields. These fields will act to focus the beam and must be accounted properly in the beam optics, especially in the low energy regions of the accelerator. We will discuss this problem in greater depth in injector lectures. Let A(x,y,) be the vector potential describing the longitudinal mode (Loren gauge) 1 A c t A A c c Graduate Accelerator Physics Fall 17

64 Graduate Accelerator Physics Fall 17...,..., 1 1 r r r A A r A For cylindrically symmetrical accelerating mode, functional form can only depend on r and Maxwell s Equations give recurrence formulas for succeeding approximations 1 1 1, 1, n n n n n n c d d n A c d A d A n

65 Gauge condition satisfied when da d n i n c in the particular case n = da d i c Electric field is 1 E c A t Graduate Accelerator Physics Fall 17

66 And the potential and vector potential must satisfy E i E c d d i c, A d, A 4A 1 d A c So the magnetic field off axis may be expressed directly in terms of the electric field on axis B i r ra E, c 1 Graduate Accelerator Physics Fall 17

67 And likewise for the radial electric field (see also ) E r r 1 r de, d Explicitly, for the time dependence cos(ωt + δ) E r E r, r,, t E, cos t, t r de, d r B sin c cos t E r,, t E, t Graduate Accelerator Physics Fall 17

68 Graduate Accelerator Physics Fall 17

69 Graduate Accelerator Physics Fall 17

70 Graduate Accelerator Physics Fall 17

71 Graduate Accelerator Physics Fall 17 Motion of a particle in this EM field B c V E e dt m d V ' ' ' sin ' ' ' ' cos ' ' ' - d t G c x t d dg x x x

72 The normalied gradient is G ee, mc and the other quantities are calculated with the integral equations G' cos t' t lim G ' ' cos t c ' d' d' c ' d' Graduate Accelerator Physics Fall 17

73 These equations may be integrated numerically using the cylindrically symmetric CEBAF field model to form the Douglas model of the cavity focussing. In the high energy limit the expressions simplify. x a x x x a a ' ' ' ' x' G' a ' ' a x d' ' cos t d' Graduate Accelerator Physics Fall 17

74 Graduate Accelerator Physics Fall 17 Transfer Matrix E E I L E E T G G For position-momentum transfer matrix d c G d c G I / sin sin / cos cos

75 Kick Generated by mis-alignment E G E Graduate Accelerator Physics Fall 17

76 Damping and Antidamping By symmetry, if electron traverses the cavity exactly on axis, there is no transverse deflection of the particle, but there is an energy increase. By conservation of transverse momentum, there must be a decrease of the phase space area. For linacs NEVER use the word adiabatic d mv transverse dt x x Graduate Accelerator Physics Fall 17

77 Conservation law applied to angles, 1 x y / / x x x y y y x y x y Graduate Accelerator Physics Fall 17

78 Phase space area transformation dx dx dx d dy dy dy d Therefore, if the beam is accelerating, the phase space area after the cavity is less than that before the cavity and if the beam is decelerating the phase space area is greater than the area before the cavity. The determinate of the transformation carrying the phase space through the cavity has determinate equal to Det M cavity x y Graduate Accelerator Physics Fall 17

79 By concatenation of the transfer matrices of all the accelerating or decelerating cavities in the recirculated linac, and by the fact that the determinate of the product of two matrices is the product of the determinates, the phase space area at each location in the linac is dx dy d x d y dx dy d x d y Same type of argument shows that things like orbit fluctuations are damped/amplified by acceleration/deceleration. Graduate Accelerator Physics Fall 17

80 Transfer Matrix Non-Unimodular M P P P tot M M 1 M M det M unimodular! M det M M det M tot 1 M PM PM tot M det M M tot 1 can separately track the"unimodular part"(as before!) and normalie by accumulated determinate 1 Graduate Accelerator Physics Fall 17

81 Bettor Phasor Diagram Off crest, s synchrotron phase Vc i Vc 1 i tan e ig i cos b s V c Vc cos e i i g i b i g, opt V tan i sin c b s Graduate Accelerator Physics Fall 17

82 Subharmonic Beam Loading Under condition of constant incident RF power, there is a voltage fluctuation in the fundamental accelerating mode when the beam load is sub harmonically related to the cavity frequency Have some old results, from the days when we investigated FELs in the CEBAF accelerator These results can be used to quantify the voltage fluctuations expected from the subharmonic beam load in LCLS II Graduate Accelerator Physics Fall 17

83 CEBAF FEL Results Bunch repetition time τ 1 chosen to emphasie physics Krafft and Laubach, CEBAF-TN-153 (1989) Graduate Accelerator Physics Fall 17

84 Model Single standing wave accelerating mode. Reflected power absorbed by matched circulator. d Vc c dvc c dv d ZI cvc dt QL dt Qc dt dt Beam current b (Constant) Incident RF (β coupler coupling) V ZP cos t I t q t l I t l l g l c b Graduate Accelerator Physics Fall 17

85 Analytic Method of Solution Green function c tt Gt t exp sin ˆ t t ˆ 11/ 4Q QL Geometric series summation c c c L c tn /QL RP crq e c / QL c /QL V cos cos ˆ cos ˆ c t ct e c t n e c t n 1 1 Q D Excellent approximation RP R ctn/ql Vc t cos ct IQLe cosct 1 Q Graduate Accelerator Physics Fall 17

86 Phasor Diagram of Solution Vc R IQL Q RP Vc 1 Graduate Accelerator Physics Fall 17

87 Single Subharmonic Beam Voltage Deviation V c R cq Q Graduate Accelerator Physics Fall 17

88 Beam Cases Case 1 Beam Beam Pulse Rep. Rate Bunch Charge (pc) Average Current (µa) Case HXR 1 MH Straight 1 kh SXR 1 MH Beam Beam Pulse Rep. Rate Bunch Charge (pc) Average Current (µa) Case 3 HXR 1 MH Straight 1 kh SXR 1 kh Beam Beam Pulse Rep. Rate Bunch Charge (pc) Average Current (µa) HXR 1 kh Straight 1 kh SXR 1 kh Graduate Accelerator Physics Fall 17

89 Case 1 V t c Straight Current HXR Current SXR Current Total Voltage µsec 1 µsec t Graduate Accelerator Physics Fall 17

90 Volts Case 1 ΔE/E= t/ nsec Graduate Accelerator Physics Fall 17

91 Volts Case ΔE/E= t/ nsec Graduate Accelerator Physics Fall 17

92 Volts Case 3 ΔE/E=1-4 t/1 nsec Graduate Accelerator Physics Fall 17

93 Summaries of Beam Energies Case 1 For off-crest cavities, multiply by cos φ Case Beam Minimum (kv) Maximum (kv) Form HXR Linear 1 Straight Constant SXR Linear 1 Beam Minimum (kv) Maximum (kv) Form HXR Linear 1 Straight Constant SXR Linear 1 Case 3 Beam Minimum (kv) Maximum (kv) Form HXR Linear 1 Straight Constant SXR Linear 1 Graduate Accelerator Physics Fall 17

94 Summary Fluctuations in voltage from constant intensity subharmonic beams can be computed analytically Basic character is a series of steps at bunch arrival, the step magnitude being (R/Q)πf c q Energy offsets were evaluated for some potential operating scenarios. Spread sheet provided that can be used to investigate differing current choices Graduate Accelerator Physics Fall 17

95 Case I V t c Zero Crossing Left Deflection Right Deflection Total Voltage µsec 1 µsec t Graduate Accelerator Physics Fall 17

96 Case (1 kh contribution minor) V t c Zero Crossing Left Deflection Right Deflection Total Voltage µsec 1 µsec t Graduate Accelerator Physics Fall 17

97 Case 3 V t c Zero Crossing Left Deflection Right Deflection Total Voltage µsec 1 µsec t Graduate Accelerator Physics Fall 17

98 RF Acceleration Characteriing Superconducting RF (SRF) Accelerating Structures Terminology Energy Gain, R/Q, Q, Q L and Q ext RF Equations and Control Coupling Ports Beam Loading RF Focusing Betatron Damping and Anti-damping Graduate Accelerator Physics Fall 17

99 Terminology 1 CEBAF Cavity 5 Cell Cavity Cells 1 DESY Cavity 9 Cell Cavity Graduate Accelerator Physics Fall 17

100 Modern Jefferson Lab Cavities (1.497 GH) are optimied around a 7 cell design Typical cell longitudinal dimension: λ RF / Phase shift between cells: π Cavities usually have, in addition to the resonant structure in picture: (1) At least 1 input coupler to feed RF into the structure () Non-fundamental high order mode (HOM) damping (3) Small output coupler for RF feedback control Graduate Accelerator Physics Fall 17

101 Graduate Accelerator Physics Fall 17

102 Some Fundamental Cavity Parameters Energy Gain d mc ee x t, t v dt For standing wave RF fields and velocity of light particles, cos,, cos / E x t E x t mc e E d RF RF i ee / RF e c.c. = V ee / Normalie by the cavity length L for gradient Vc Eacc MV/m L c RF Graduate Accelerator Physics Fall 17