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Accelerator Physics Particle Acceleration S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage Old Dominion University Jefferson Lab Lecture 6 USPAS Accelerator Physics June 2016 RF Acceleration • • • • Characterizing Superconducting RF (SRF) Accelerating Structures – Terminology – Energy Gain, R/Q, Q0, QL and Qext RF Equations and Control – Coupling Ports – Beam Loading RF Focusing Betatron Damping and Anti-damping USPAS Accelerator Physics June 2016 Terminology 5 Cell Cavity 1 CEBAF Cavity “Cells” 9 Cell Cavity 1 DESY Cavity USPAS Accelerator Physics June 2016 Modern Jefferson Lab Cavities (1.497 GHz) are optimized around a 7 cell design Typical cell longitudinal dimension: λRF/2 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 (2) Non-fundamental high order mode (HOM) damping (3) Small output coupler for RF feedback control USPAS Accelerator Physics June 2016 USPAS Accelerator Physics June 2016 Some Fundamental Cavity Parameters • Energy Gain d mc 2 eE x t , t v dt • For standing wave RF fields and velocity of light particles E x , t E x cos RF t mc 2 e E 0, 0, z cos 2 z / z RF = eEz 2 / RF e i c.c. Vc eEz 2 / RF 2 • Normalize by the cavity length L for gradient E acc MV/m Vc L USPAS Accelerator Physics June 2016 dz Shunt Impedance R/Q • Ratio between the square of the maximum voltage delivered by a cavity and the product of ωRF and the energy stored in a cavity (using “accelerator” definition) Vc2 R Q RF stored energy • Depends only on the cavity geometry, independent of frequency when uniformly scale structure in 3D • Piel’s rule: R/Q ~100 Ω/cell CEBAF 5 Cell 480 Ω CEBAF 7 Cell 760 Ω DESY 9 Cell 1051 Ω USPAS Accelerator Physics June 2016 Unloaded Quality Factor • As is usual in damped harmonic motion define a quality factor by Q 2 energy stored in oscillation energy dissipated in 1 cycle • Unloaded Quality Factor Q0 of a cavity Q0 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. USPAS Accelerator Physics June 2016 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 Q0 • Coupling Factor Q0 Q0 1 for present day SRF cavities, QL Qext 1 • 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 minimize operating RF power: current matching (CEBAF, FEL), rf control performance and microphonics (SNS, ERLs) USPAS Accelerator Physics June 2016 Q0 vs. Gradient for Several 1300 MHz Cavities Q0 1011 1010 AC70 AC72 AC73 AC78 AC76 AC71 AC81 Z83 Z87 Courtesey: Lutz Lilje 109 0 10 20 Eacc [MV/m] USPAS Accelerator Physics June 2016 30 40 Eacc vs. time 45 40 35 BCP EP 10 per. Mov. Avg. (BCP) 10 per. Mov. Avg. (EP) Eacc[MV/m] 30 25 20 15 10 5 Courtesey: Lutz Lilje 0 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 USPAS Accelerator Physics June 2016 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 Qext Optimization under Beam Loading and Microphonics RF Modeling USPAS Accelerator Physics June 2016 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 analyze it We will write the quantities that characterize 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 USPAS Accelerator Physics June 2016 RF Cavity Fundamental Quantities Quality Factor Q0: Q0 0W Pdiss Energy stored in cavity Energy dissipated in cavity walls per radian Shunt impedance Ra (accelerator definition); Vc2 Ra Pdiss Note: Voltages and currents will be represented as complex quantities, denoted by a tilde. For example: Vc t Re Vc t eit where Vc Vc varying phase. Vc t Vc ei t is the magnitude of Vc USPAS Accelerator Physics June 2016 and is a slowly Equivalent Circuit for an rf Cavity Simple LC circuit representing an accelerating resonator. Metamorphosis of the LC circuit into an accelerating cavity. Chain of weakly coupled pillbox cavities representing an accelerating cavity. Chain of coupled pendula as its mechanical analogue. USPAS Accelerator Physics June 2016 Equivalent Circuit for an RF Cavity An rf cavity can be represented by a parallel LCR circuit: V c (t ) V c eit Impedance Z of the equivalent circuit: Resonant frequency of the circuit: Stored energy W: 1 W CVc2 2 USPAS Accelerator Physics June 2016 1 1 Z iC R iL 0 1/ LC 1 Equivalent Circuit for an RF Cavity Average power dissipated in resistor R: Vc2 Pdiss 2R From definition of shunt impedance Vc2 Ra Pdiss Ra 2 R Quality factor of resonator: Q0 0W Pdiss 0 Note: Z R 1 iQ0 0 1 0CR For 0 , 0 Z R 1 2iQ0 0 USPAS Accelerator Physics June 2016 1 Wiedemann 19.11 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 USPAS Accelerator Physics June 2016 Cavity with External Coupling Consider the rf cavity after the rf is turned off. Stored energy W satisfies the equation: dW / dt Ptot Total power being lost, Ptot, is: P P P P tot diss e t Pe is the power leaking back out the input coupler. Pt is the power coming out the transmitted power coupler. Typically Pt is very small Ptot Pdiss + Pe Recall Q0 0W Pdiss QL 0W Ptot t 0 dW 0W W W0e QL dt QL Energy in the cavity decays exponentially with time constant: L QL / 0 USPAS Accelerator Physics June 2016 Decay rate equation Ptot Pdiss Pe 0W 0W suggests that we can assign a quality factor to each loss mechanism, such that 1 1 1 QL Q0 Qe where, by definition, Qe 0W Pe Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107. USPAS Accelerator Physics June 2016 Have defined “coupling parameter”: and therefore Pe Q 0 Pdiss Qe 1 (1 ) QL Q0 Wiedemann 19.9 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. USPAS Accelerator Physics June 2016 Cavity Coupled to an RF Source The system we want to model. A generator producing power Pg transmits power to cavity through transmission line with characteristic impedance Z0 Z0 Z0 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. USPAS Accelerator Physics June 2016 Transmission Lines I n2 L Vn 1 C I n 1 L Vn C … In L VN I N C Inductor Impedance and Current Conservation Vn 1 Vn i LI n 1 I n 1 I n iCVn USPAS Accelerator Physics June 2016 Z Transmission Line Equations • Standard Difference Equation with Solution Vn V0 e in , I n I 0e in V0 ei 1 i LI 0ei I 0 ei 1 iCV0 2sin /2 LC V L / C I ei /2 V L / C I ei /2 • Continuous Limit ( N → ∞) LC k LC V x, t V0 eit kx v 1/ LC L / C I 0 eit kx Z 0 I 0 eit kx V x, t V0 eit kx L / C I 0eit kx Z 0 I 0eit kx USPAS Accelerator Physics June 2016 Cavity Coupled to an RF Source Equivalent Circuit Z0 ik I I V , I V , I 1: k RF Generator + Circulator Coupler Cavity Coupling is represented by an ideal (lossless) transformer of turns ratio 1:k USPAS Accelerator Physics June 2016 Cavity Coupled to an RF Source From transmission line equations, forward power from RF source V 2 / 2Z 0 Reflected power to circulator V 2 / 2Z 0 Transformer relations Vc kVk k V V ic ik / k I I / k 2 I / k Vc / k 2 Z 0 Considering zero forward power case and definition of β V 2 / 2Z0 / Vc 2 / 2R R / k 2 Z0 USPAS Accelerator Physics June 2016 Cavity Coupled to an RF Source Loaded cavity looks like ig (t ) 2I t Wiedemann Fig. 19.1 k Re ig eit R R R 2 Zg Z g k Z0 Effective and loaded resistance 1 1 1 1 Reff R Z g R Wiedemann 19.1 Ra RL 2 Reff 1 Solving transmission line equations 1 Vc V kZ 0ic 2 k 1 Vc V kZ 0ic 2 k USPAS Accelerator Physics June 2016 Powers Calculated Forward Power 0 V Vc2 1 1 Pg 1 1 iQ0 2 8Z 0 k Ra 4 0 2 2 c 2 2 0 2 1 QL 0 Reflected Power 0 V Vc2 1 1 Prefl 1 1 iQ0 2 8Z 0 k Ra 4 0 2 2 c 2 2 12 0 2 QL 2 1 0 Power delivered to cavity is V 1 Pg Prefl Ra 4 2 c 2 12 V 2 c Pdiss 1 2 1 Ra as it must by energy conservation! USPAS Accelerator Physics June 2016 Some Useful Expressions Total energy W, in terms of cavity parameters Q0 P 0 diss W 4 Pg Pdiss 1 2 W 4 0 W Q0 0 Total impedance 1 1 Q 0 0 1 2 2 L Pg 2 0 (1 ) 2 Q02 0 Q0 4 1 Pg 2 (1 ) 2 0 0 1 2QL 0 ZTOT 1 1 Z Z g ZTOT R a 2 1 0 (1 ) iQ 0 0 USPAS Accelerator Physics June 2016 1 When Cavity is Detuned Define “Tuning angle” : 0 tan QL 0 2QL 0 0 4 Q0 1 W= Pg 2 2 (1+ ) 0 1+tan Note that: Pdiss 4 1 Pg 2 (1 ) 1 tan 2 USPAS Accelerator Physics June 2016 for 0 Wiedemann 19.13 Optimal β Without Beam . . Optimal coupling: W/Pg maximum or Pdiss = Pg which implies for Δω = 0, β = 1 This is the case called “critical” coupling Reflected power is consistent with energy conservation: Prefl Pg Pdiss 4 1 Prefl Pg 1 2 2 (1 ) 1 tan . Pdiss 1 0.9 0.8 On resonance: W Dissipated and Reflected Power 4 Q0 Pg (1 ) 2 0 4 Pg 2 (1 ) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 1 Prefl Pg 1 0 1 USPAS Accelerator Physics June 2016 2 3 4 5 6 7 8 9 10 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: iC C dVc , dt iR Vc , RL / 2 Vc L diL dt ig the current induced by generator, ib beam current Differential equation that describes the dynamics of the system: d 2Vc 0 dVc 0 RL d 2 V ig ib 0 c 2 dt QL dt 2QL dt USPAS Accelerator Physics June 2016 Cavity with Beam Kirchoff’s law: 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 Vc t Re Vc eit ig t Re ig eit ib t Re ib eit where is the generator angular frequency and are complex quantities. USPAS Accelerator Physics June 2016 Vc , ig , ib Voltage for a Cavity with Beam Steady state solution RL (1 i tan )Vc (ig ib ) 2 where is the tuning angle. Generator current 2 Pg 2 2 Pg | ig | 2 I 2 4 k Z0 R For short bunches: beam current. | ib | 2 I 0 Pg Ra where I0 is the average Wiedemann 19.19 USPAS Accelerator Physics June 2016 Voltage for a Cavity with Beam At steady-state: or RL / 2 RL / 2 ig ib (1 i tan ) (1 i tan ) R R Vc L ig cos ei L ib cos ei 2 2 Vc or Vc Vgr cos ei Vbr cos ei or Vc RL V i g gr 2 R L V ib br 2 Vg Vb are the generator and beam-loading voltages on resonance and Vg Vb are the generator and beam-loading voltages. USPAS Accelerator Physics June 2016 Voltage for a Cavity with Beam Note that: | Vgr | 2 1 Pg RL 2 Pg RL for large | Vbr | RL I 0 Wiedemann 19.16 Wiedemann 19.20 USPAS Accelerator Physics June 2016 Voltage for a Cavity with Beam Im(Vg ) Vg Vgr cos e i Vgr cos ei Vb Vbr cos ei Re(Vg ) Vgr As increases, the magnitudes of both Vg and Vb decrease while their phases rotate by . USPAS Accelerator Physics June 2016 Example of a Phasor Diagram Vb Vbr Vg b I acc Vc Ib I dec Wiedemann Fig. 19.3 USPAS Accelerator Physics June 2016 On Crest/On Resonance Operation Typically linacs operate on resonance and on crest in order to receive maximum acceleration. On crest and on resonance Ib Vbr Vc Vc Vgr Vbr where Vc is the accelerating voltage. USPAS Accelerator Physics June 2016 Vgr More Useful Equations We derive expressions for W, Va, Pdiss, Prefl in terms of and the loading parameter K, defined by: K=I0Ra/(2 Pg ) 2 Vc Pg RL 1 From: | Vgr | 2 1 | Vbr | RL I 0 Vc Vgr Vbr K 1 2 4 Q0 K W 1 Pg (1 ) 2 0 Pg RL 2 Pdiss 4 K 1 Pg (1 ) 2 I 0Va I 0 Ra Pdiss 2 I 0Vc K 2 K 1 Pg 1 2 Prefl Pg Pdiss I 0Va Prefl USPAS Accelerator Physics June 2016 ( 1) 2 K P g 2 ( 1) Clarifications On Crest, Vc 2 1 Pg RL RL I 0 RL I 0 1 Pg RL 1 Pg RL 2 2 1 Ra I 0 K 2 Pg Off Crest with Detuning Pg Z0 I 2 2 2 = Z 0 k ig 2 8 2Vc ig 1 i tan 2 I 0 cos b i sin b RL 2 Pg Z 0 k Vc 2 RL2 2 2 2 I R I R 1 0 L cos b tan 0 L sin b Vc Vc USPAS Accelerator Physics June 2016 More Useful Equations For large, Pg 1 (Vc I 0 RL ) 2 4 RL Prefl 1 (Vc I 0 RL ) 2 4 RL For Prefl=0 (condition for matching) VcM RL M I0 and Pg M 0 M c I V 4 Vc I0 M M V I 0 c USPAS Accelerator Physics June 2016 2 Example For Vc=14 MV, L=0.7 m, QL=2x107 , Q0=1x1010 : I0 = 0 I0 = 100 A I0 = 1 mA 3.65 kW 4.38 kW 14.033 kW Pdiss 29 W 29 W 29 W I0Vc 0W 1.4 kW 14 kW Prefl 3.62 kW 2.951 kW ~ 4.4 W Power Pg USPAS Accelerator Physics June 2016 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 minimize 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 b 2 I 0ei b Vc Vcei c and set c 0 The generator power can then be expressed as: 2 2 Vc2 (1 ) I 0 RL I 0 RL Pg 1 cos tan sin b b RL 4 Vc Vc USPAS Accelerator Physics June 2016 Wiedemann 19.30 Off Crest/Off Resonance Operation Condition for optimum tuning: tan I 0 RL sin b Vc Condition for optimum coupling: opt 1 I 0 Ra cos b Vc Minimum generator power: Pg ,min Vc2 opt Ra USPAS Accelerator Physics June 2016 Wiedemann 19.36 Bettor Phasor Diagram Off crest, s synchrotron phase Vc 1 i tan Vc ei ig ib cos s Vc Vc i e cos ig ib USPAS Accelerator Physics June 2016 ig ,opt Vc tan ib sin s C75 Power Estimates G. A. Krafft USPAS Accelerator Physics June 2016 12 GeV Project Specs 7-cell, 1500 MHz, 903 ohms 20 18 16 13 kW 14 P (kW) 12 10 8 Delayen and Krafft TN-07-29 21.2 MV/m, 460 uA, 25 Hz, 0 deg 6 21.2 MV/m, 460 uA, 20 Hz, 0 deg 4 21.2 MV/m, 460 uA, 15 Hz, 0 deg 21.2 MV/m, 460 uA, 10 Hz, 0 deg 2 21.2 MV/m, 460 uA, 0 Hz, 0 deg 0 1 10 100 Qext (106) 460 µA*15 MV=6.8 kW USPAS Accelerator Physics June 2016 7-cell, 1500 MHz, 903 ohms 20 18 16 P (kW) 14 12 10 8 25.0 MV/m, 460 uA, 25 Hz, 0 deg 6 24.0 MV/m, 460 uA, 25 Hz, 0 deg 4 23.0 MV/m, 460 uA, 25 Hz, 0 deg 22.0 MV/m, 460 uA, 25 Hz, 0 deg 2 21.0 MV/m, 460 uA, 25 Hz, 0 deg 0 1 10 100 6 Qext (10 ) USPAS Accelerator Physics June 2016 Assumptions • Low Loss R/Q = 903*5/7 = 645 Ω • Max Current to be accelerated 460 µA • Compute 0 and 25 Hz detuning power curves • 75 MV/cryomodule (18.75 MV/m) • Therefore matched power is 4.3 kW (Scale increase 7.4 kW tube spec) • Qext adjustable to 3.18×107(if not need more RF power!) USPAS Accelerator Physics June 2016 Power vs. Qext 14000 12000 25 Hz detuning 10000 Pg (kW) 8000 6000 4000 3.2×107 0 detuning 2000 0 1 10 Qext (×106) USPAS Accelerator Physics June 2016 100 RF Cavity with Beam and Microphonics The detuning is now: tan 2QL f0 fm tan 0 2QL f0 f0 f0 where f0 is the static detuning (controllable) Probability Density Medium CM Prototype, Cavity #2, CW @ 6MV/m 400000 samples 10 8 6 4 0.25 2 0 -2 -4 -6 -8 -10 90 95 100 105 Time (sec) 110 115 120 Probability Density Frequency (Hz) and f m is the random dynamic detuning (uncontrollable) 0.2 0.15 0.1 0.05 0 -8 -6 -4 -2 0 2 Peak Frequency Deviation (V) USPAS Accelerator Physics June 2016 4 6 8 Qext Optimization with Microphonics 2 2 V (1 ) Itot RL Itot RL Pg 1 cos tan sin tot tot RL 4 Vc Vc 2 c tan 2QL f f0 where f is the total amount of cavity detuning in Hz, including static detuning and microphonics. Optimizing the generator power with respect to coupling gives: f opt (b 1)2 2Q0 b tan tot f0 I R where b tot a cos tot Vc 2 where Itot 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. USPAS Accelerator Physics June 2016 Correct Static Detuning To minimize generator power with respect to tuning: f0 f0 b tan tot 2Q0 2 Vc2 1 f 2 m Pg (1 b ) 2Q0 Ra 4 f 0 The resulting power is Vc2 1 Pg Ra 4 opt 2 f 2 2 m 1 b 2 1 b opt opt 2Q0 f 0 Vc2 1 b opt 2 Ra USPAS Accelerator Physics June 2016 Optimal Qext and Power Condition for optimum coupling: 2 opt fm 2 (b 1) 2Q0 f 0 Pgopt 2 Vc2 f b 1 (b 1) 2 2Q0 m 2 Ra f0 and In the absence of beam (b=0): and 2 opt fm 1 2Q0 f 0 Pgopt 2 V f 1 1 2Q0 m 2 Ra f0 2 c USPAS Accelerator Physics June 2016 Problem for the Reader Assuming no microphonics, plot opt and Pgopt as function of b (beam loading), b=-5 to 5, and explain the results. How do the results change if microphonics is present? USPAS Accelerator Physics June 2016 Example ERL Injector and Linac: fm=25 Hz, Q0=1x1010 , f0=1300 MHz, I0=100 mA, Vc=20 MV/m, L=1.04 m, Ra/Q0=1036 ohms per cavity ERL linac: Resultant beam current, Itot = 0 mA (energy recovery) and opt=385 QL=2.6x107 Pg = 4 kW per cavity. ERL Injector: I0=100 mA and opt= 5x104 ! QL= 2x105 Pg = 2.08 MW per cavity! Note: I0Va = 2.08 MW optimization is entirely dominated by beam loading. USPAS Accelerator Physics June 2016 RF System Modeling To include amplitude and phase feedback, nonlinear effects from the klystron and be able to analyze 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, Lorentz force detuning and coupling and excitation of mechanical resonances USPAS Accelerator Physics June 2016 RF System Model USPAS Accelerator Physics June 2016 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, 100 sec beam pulse enters the cavity. USPAS Accelerator Physics June 2016 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 analyze steady-state situations. We presented formula for the optimization of Qext 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. USPAS Accelerator Physics June 2016 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,z) be the vector potential describing the longitudinal mode (Lorenz gauge) 1 A c t 2 2 2 2 A 2 A 2 c c USPAS Accelerator Physics June 2016 For cylindrically symmetrical accelerating mode, functional form can only depend on r and z Az r , z Az 0 z Az1 z r 2 ... r , z 0 z 1 z r 2 ... Maxwell’s Equations give recurrence formulas for succeeding approximations 2 2n2 Azn d Az ,n1 2 2 2 Az ,n1 dz c 2 2 d 2 2n n n21 2 n1 dz c USPAS Accelerator Physics June 2016 Gauge condition satisfied when dAzn i n dz c in the particular case n = 0 dAz 0 i 0 dz c Electric field is 1 A E c t USPAS Accelerator Physics June 2016 And the potential and vector potential must satisfy d0 i E z 0, z Az 0 dz c d Az 0 i Ez 0, z 2 Az 0 4 Az1 2 c dz c 2 2 So the magnetic field off axis may be expressed directly in terms of the electric field on axis i r B 2rAz1 E z 0, z 2 c USPAS Accelerator Physics June 2016 And likewise for the radial electric field (see also E 0) r dE z 0, z Er 2r1 z 2 dz Explicitly, for the time dependence cos(ωt + δ) Ez r, z, t Ez 0, z cost r dE z 0, z Er r , z , t cost 2 dz B r , z , t r 2c E z 0, z sin t USPAS Accelerator Physics June 2016 USPAS Accelerator Physics June 2016 USPAS Accelerator Physics June 2016 USPAS Accelerator Physics June 2016 Motion of a particle in this EM field V d mV e E B dt c z x z x xz ' dG z ' cost z ' z dz ' 2 dz ' z z 'xz ' z z ' - G z 'sin t z ' 2c USPAS Accelerator Physics June 2016 The normalized gradient is eE z z ,0 Gz mc2 and the other quantities are calculated with the integral equations z z G z ' cost z ' dz ' G z ' z z z z cost z ' dz ' z ' z z z z0 dz ' t z lim z0 c z 'c z z USPAS Accelerator Physics June 2016 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. z ' x z ' xz xa dz ' z ' z z ' a z x x z ' G z ' z a xa cost z ' dz ' 2 z 2 z ' z z ' a z USPAS Accelerator Physics June 2016 Transfer Matrix For position-momentum transfer matrix L EG 1 2E T EG I 4 1 2 E I cos 2 G 2 z cos 2 z / c dz sin 2 G 2 z sin 2 z / c dz USPAS Accelerator Physics June 2016 Kick Generated by mis-alignment EG 2E USPAS Accelerator Physics June 2016 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 mVtransverse 0 dt z x z x USPAS Accelerator Physics June 2016 Conservation law applied to angles x , y z 1 x x / z x y y / z y z x z x z z z z y z y z z z USPAS Accelerator Physics June 2016 Phase space area transformation z dx d x dx d x z z z z z dy d y dy d y z z z z 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 z Det M cavity z z z USPAS Accelerator Physics June 2016 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 0 z 0 dx d x z dx d x 0 z z z 0 z 0 dy d y z dy d y 0 z z z Same type of argument shows that things like orbit fluctuations are damped/amplified by acceleration/deceleration. USPAS Accelerator Physics June 2016 Transfer Matrix Non-Unimodular M tot M 1 M 2 M PM det M PM unimodular ! M tot M1 M2 PM tot PM 1 PM 2 det M tot det M 1 det M 2 can separately track the " unimodular part" (as before! ) and normalize by accumulate d determinat e USPAS Accelerator Physics June 2016 LCLS II Subharmonic Beam Loading USPAS Accelerator Physics June 2016 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 USPAS Accelerator Physics June 2016 CEBAF FEL Results Bunch repetition time τ1 chosen to emphasize physics Krafft and Laubach, CEBAF-TN-0153 (1989) USPAS Accelerator Physics June 2016 Model • Single standing wave accelerating mode. Reflected power absorbed by matched circulator. d 2Vc c dVc c dV d ZIb 2 c Vc 2 dt QL dt Qc dt dt • Beam current Ib t l l q t l I t l • (Constant) Incident RF (β coupler coupling) V 2 2ZPg cos ct USPAS Accelerator Physics June 2016 Analytic Method of Solution • Green function c t t G t t exp sin ˆ c t t 2QL • Geometric series summation ˆ c c 1 1/ 4QL2 2 RP c Rq ec t n /2QL c / QL c /2 QL ˆ ˆ Vc t cos ct e cos t n e cos t n 1 c c 1 2Q D • Excellent approximation 2 RP R Vc t cos ct IQL ec t n /2QL cos ct 1 Q USPAS Accelerator Physics June 2016 Phasor Diagram of Solution R IQL Q Vc 2 RP 1 USPAS Accelerator Physics June 2016 Vc Single Subharmonic Beam 1.2 Voltage Deviation 1 1 Vc R c q Q 2 0.8 0.6 0.4 0.2 0 0 200 400 600 USPAS Accelerator Physics June 2016 800 1000 1200 Beam Cases • Case 1 Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA) HXR 1 MHz 145 145 Straight 10 kHz 145 1.45 SXR 1 MHz 145 145 • Case 2 Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA) 1 MHz 10 kHz 100 kHz 295 295 20 295 2.95 2 Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µA) HXR 100 kHz 295 295 Straight 10 kHz 295 2.95 SXR 100 kHz 20 2 HXR Straight SXR • Case 3 Beam USPAS Accelerator Physics June 2016 Case 1 Straight Current HXR Current SXR Current Total Voltage Vc t 2 µsec t 100 µsec USPAS Accelerator Physics June 2016 Case 1 Volts ΔE/E=10-4 ... t/20 nsec USPAS Accelerator Physics June 2016 Volts Case 2 ΔE/E=10-4 ... t/20 nsec USPAS Accelerator Physics June 2016 Case 3 Volts ΔE/E=10-4 t/100 nsec USPAS Accelerator Physics June 2016 Summaries of Beam Energies For off-crest cavities, multiply by cos φ • Case 1 Beam HXR Straight SXR Minimum (kV) Maximum (kV) Form 0.615 0.908 0.618 1.222 0.908 1.225 Linear 100 Constant Linear 100 Minimum (kV) Maximum (kV) Form 0.631 1.550 0.788 1.943 1.550 1.911 Linear 100 Constant Linear 10 Minimum (kV) Maximum (kV) Form 0.690 1.574 0.753 1.814 1.574 1.876 Linear 10 Constant Linear 10 • Case 2 Beam HXR Straight SXR • Case 3 Beam HXR Straight SXR USPAS Accelerator Physics June 2016 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)πfcq • Energy offsets were evaluated for some potential operating scenarios. Spread sheet provided that can be used to investigate differing current choices USPAS Accelerator Physics June 2016 Case I’ Zero Crossing Left Deflection Right Deflection Total Voltage Vc t 20 µsec t 100 µsec USPAS Accelerator Physics June 2016 Case 2 (100 kHz contribution minor) Zero Crossing Left Deflection Right Deflection Total Voltage Vc t 2 µsec t 100 µsec USPAS Accelerator Physics June 2016 Case 3 Zero Crossing Left Deflection Right Deflection Total Voltage Vc t 20 µsec t 100 µsec USPAS Accelerator Physics June 2016