Report

Accelerator Physics Particle Acceleration G. A. Krafft, Alex Bogacz and Timofey Zolkin Jefferson Lab Colorado State University Lecture 6 USPAS Accelerator Physics June 2013 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 2013 Terminology 5 Cell Cavity 1 CEBAF Cavity “Cells” 9 Cell Cavity 1 DESY Cavity USPAS Accelerator Physics June 2013 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 2013 USPAS Accelerator Physics June 2013 Some Fundamental Cavity Parameters • Energy Gain d mc2 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 Ez 0, 0, z cos z / RF dz = eEz 2 / RF e i c.c. Vc eEz 2 / RF 2 • Normalize by the cavity length L for gradient Vc E acc MV/m L USPAS Accelerator Physics June 2013 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 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 2013 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 2013 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 2013 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 2013 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 2013 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 Conclusions USPAS Accelerator Physics June 2013 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 USPAS Accelerator Physics June 2013 RF Cavity Fundamental Quantities Quality Factor Q0: Q0 0W Pdiss Energy stored in cavity Energy dissipated in cavity walls per radian Shunt impedance Ra: Va2 Ra Pdiss in ohms per cell (accelerator definition); Va = accelerating voltage 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 2013 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 2013 Equivalent Circuit for an rf Cavity An rf cavity can be represented by a parallel LCR circuit: V c (t ) vc eit Impedance Z of the equivalent circuit: Z 1 1 iC R iL Resonant frequency of the circuit: 1 W CVc2 2 Stored energy W: USPAS Accelerator Physics June 2013 0 1/ LC 1 Equivalent Circuit for an rf Cavity Power dissipated in resistor R: Pdiss 1 Vc2 2 R Ra 2 R Va2 Ra Pdiss From definition of shunt impedance Q0 0W Pdiss 0CR Quality factor of resonator: Note: Z R 1 iQ0 0 0 Wiedemann 16.13 1 For 0 , USPAS Accelerator Physics June 2013 0 Z R 1 2iQ0 0 1 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 2013 Cavity with External Coupling (cont’d) Consider the rf cavity after the rf is turned off. dW Stored energy W satisfies the equation: 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 W Q0 Recall 0 Pdiss t 0 dW 0W W W0e QL dt QL W Similarly define a “loaded” quality factor QL: QL 0 Ptot Energy in the cavity decays exponentially with time constant: Q L L 0 USPAS Accelerator Physics June 2013 Cavity with External Coupling (cont’d) 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 2013 Cavity with External Coupling (cont’d) Define “coupling parameter”: therefore is equal to: Q0 Qe 1 (1 ) QL Q0 Wiedemann 16.9 Pe Pdiss 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 2013 Cavity Coupled to an rf Source The system we want to model: Between the rf generator and the cavity is an isolator – a circulator connected to a load. Circulator ensures that signals coming from the cavity are terminated in a matched load. Equivalent circuit: I k (t ) i k eit RF Generator + Circulator Coupler Cavity Coupling is represented by an ideal transformer of turn ratio 1:k USPAS Accelerator Physics June 2013 Cavity Coupled to an rf Source I k (t ) i k eit Wiedemann Fig. 16.1 I g (t ) ig eit By definition, Ik Ig k Z g k 2Z0 R R R 2 Zg Z g k Z0 USPAS Accelerator Physics June 2013 Wiedemann 16.1 Generator Power When the cavity is matched to the input circuit, the power dissipation in the cavity is maximized. I g (t ) ig eit max diss P Ig 1 Zg 2 2 2 max diss or P 1 Ra I g2 Pg 16 Wiedemann 16.6 We define the available generator power Pg at a given generator current I g to be equal to Pdissmax . USPAS Accelerator Physics June 2013 Some Useful Expressions We derive expressions for W, Pdiss, Prefl, in terms of cavity parameters Q Q V 0 W Pg 0 Pdiss 1 Ra I g2 16 Vc I g ZTOT ZTOT 0 1 1 Z Z g 2 c 0 Ra 1 Ra I g2 16 16 Q0 Vc2 Ra2 0 I g2 1 1 0 ZTOT (1 ) iQ 0 0 Q 1 W 4 0 Pg 2 0 (1 ) 2 Q02 0 0 For 0 R a 2 W 4 Q0 2 (1 ) 0 1 Q0 0 1 2 (1 ) 0 USPAS Accelerator Physics June 2013 2 Pg Some Useful Expressions (cont’d) W 4 Q0 2 (1 ) 0 1 Q0 0 1 2 (1 ) 0 2 Pg Define “Tuning angle” : 0 tan QL 0 2QL 0 0 W= 4 Q0 1 Pg 2 2 (1+ ) 0 1+tan Recall: Pdiss 0W Q0 Pdiss 4 1 Pg (1 ) 2 1 tan 2 USPAS Accelerator Physics June 2013 for 0 Wiedemann 16.12 Some Useful Expressions (cont’d) . . Optimal coupling: W/Pg maximum or Pdiss = Pg which implies Δω = 0, β = 1 this is the case of critical coupling Reflected power is calculated from energy conservation: Prefl Pg Pdiss 4 1 Prefl Pg 1 2 2 (1 ) 1 tan . 1 0.9 0.8 On resonance: 4 Q0 W Pg (1 ) 2 0 Pdiss Dissipated and Reflected Power 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 2013 2 3 4 5 6 7 8 9 10 Equivalent Circuit for a Cavity with Beam in the rf cavity isBeam represented by a current generator. Equivalent circuit: iC C dvC , dt iR vC , RL / 2 vC L diL dt Differential equation that describes the dynamics of the system: RL is the loaded impedance defined as: USPAS Accelerator Physics June 2013 RL Ra (1 ) Equivalent Circuit for a Cavity with Beam (cont’d) 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. d 2vc 0 dvc 0 RL d 2 0 vc ig ib 2 dt QL dt 2QL dt Assume that vc , ig , ib have a fast (rf) time-varying component and a slow varying component: vc Vc eit ig I g eit ib I b eit where is the generator angular frequency and are complex quantities. USPAS Accelerator Physics June 2013 Vc , I g , Ib Equivalent Circuit for a Cavity with Beam (cont’d) Neglecting terms of order arrive at: d 2Vc dI 1 dVc , , 2 dt dt QL dt we dVc 0 0 RL (1 i tan )Vc ( I g Ib ) dt 2QL 4QL where is the tuning angle. For short bunches: beam current. | Ib | 2I 0 where I0 is the average Wiedemann 16.19 USPAS Accelerator Physics June 2013 Equivalent Circuit for a Cavity with Beam (cont’d) dVc 0 R (1 i tan )Vc 0 L ( I g Ib ) dt 2QL 4QL At steady-state: or RL / 2 RL / 2 Ig Ib (1 i tan ) (1 i tan ) R R Vc L I g cos ei L I b 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 2013 Equivalent Circuit for a Cavity with Beam (cont’d) Note that: | Vgr | 2 1 Pg RL 2 Pg RL for large | Vbr | RL I 0 Wiedemann 16.16 Wiedemann 16.20 USPAS Accelerator Physics June 2013 Equivalent Circuit for a Cavity with Im(VBeam ) g Vg Vgr cos e i Vgr cos ei Vb Vbr cos ei Re(Vg ) Vgr As increases the magnitude of both Vg and Vb decreases while their phases rotate by . USPAS Accelerator Physics June 2013 Equivalent Circuit for a Cavity with Beam (cont’d) Vc Vg Vb Cavity voltage is the superposition of the generator and beam-loading voltage. This is the basis for the vector diagram analysis. USPAS Accelerator Physics June 2013 Example of a Phasor Diagram Vb Vbr Vg b I acc Vc Ib I dec Wiedemann Fig. 16.3 USPAS Accelerator Physics June 2013 On Crest and 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 Va Vgr Vbr where Va is the accelerating voltage. USPAS Accelerator Physics June 2013 Vgr More Useful Equations We derive expressions for W, Va, Pdiss, Prefl in terms of and the loading parameter K, defined by: K=I0/2 Ra/Pg 2 Va Pg Ra 1 From: | Vgr | 2 1 | Vbr | RL I 0 Va 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 0Va K 2 K 1 Pg 1 2 Prefl Pg Pdiss I 0Va Prefl USPAS Accelerator Physics June 2013 ( 1) 2 K P g 2 ( 1) More Useful Equations (cont’d) For large, Pg 1 (Va I 0 RL ) 2 4 RL Prefl 1 (Va I 0 RL ) 2 4 RL For Prefl=0 (condition for matching) VaM RL M I0 and Pg M 0 M a I V 4 Va I0 M M V I 0 a USPAS Accelerator Physics June 2013 2 Example For Va=20 MV/m, 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 I0Va 0W 1.4 kW 14 kW Prefl 3.62 kW 2.951 kW ~ 4.4 W Power Pg USPAS Accelerator Physics June 2013 Off Crest and 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 operation in recirculating linacs. We write the beam current and the cavity voltage as Ib 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 2013 Wiedemann 16.31 Off Crest and 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 2013 Wiedemann 16.36 RF Cavity with Beam and Microphonics The detuning is now: tan 2QL f0 f m 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 2013 4 6 8 Qext Optimization under Beam Loading and Microphonics Beam loading and microphonics require careful optimization of the external Q of cavities. Derive expressions for the optimum setting of cavity parameters when operating under a) heavy beam loading b) little or no beam loading, as is the case in energy recovery linac cavities and in the presence of microphonics. USPAS Accelerator Physics June 2013 Qext Optimization (cont’d) 2 2 V (1 ) Itot RL I tot 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 2 opt (b 1) 2Q0 b tan tot f 0 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 2013 Qext Optimization (cont’d) 2 2 V (1 ) Itot RL I tot RL Pg cos tot tan sin tot 1 RL 4 Vc Vc 2 c tan 2QL where: f0 fm f0 To minimize generator power with respect to tuning: f0 f0 b tan 2Q0 2 Vc2 (1 ) f 2 m Pg (1 b ) 2Q0 RL 4 f 0 USPAS Accelerator Physics June 2013 Qext Optimization (cont’d) 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 2013 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 2013 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 2013 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 2013 RF System Model USPAS Accelerator Physics June 2013 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 2013 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 2013 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 2013 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 2n 2 Azn d Az ,n 1 2 2 2 Az ,n 1 dz c 2 2 d 2 2n n n21 2 n1 dz c USPAS Accelerator Physics June 2013 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 2013 And the potential and vector potential must satisfy d 0 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 2013 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 cos t 2 dz B r , z , t r 2c E z 0, z sin t USPAS Accelerator Physics June 2013 USPAS Accelerator Physics June 2013 USPAS Accelerator Physics June 2013 USPAS Accelerator Physics June 2013 Motion of a particle in this EM field V d mV e E B dt c z x z x xz ' dGz ' cost z ' z dz' 2 dz' z z 'xz ' z z ' - G z 'sin t z ' 2c USPAS Accelerator Physics June 2013 The normalized gradient is eE z z ,0 G z 2 mc and the other quantities are calculated with the integral equations z z Gz'cost z' dz' Gz' 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 2013 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 2013 Transfer Matrix For position-momentum transfer matrix L EG 1 2E T EG I 4 1 2 E I cos2 G 2 z cos2 z / c dz sin 2 G 2 z sin 2 z / c dz USPAS Accelerator Physics June 2013 Kick Generated by mis-alignment E G 2E USPAS Accelerator Physics June 2013 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 2013 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 2013 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 2013 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 2013 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 separatelytrack the" unimodular part"(as before!) and normalizeby accumulated determinate USPAS Accelerator Physics June 2013 ENERGY RECOVERY WORKS Gradient modulator drive signal in a linac cavity measured without energy recovery (signal level around 2 V) and with energy recovery (signal level around 0). GASK 2.5 2 Voltage (V) 1.5 1 0.5 0 -1.00E-04 0.00E+00 1.00E-04 2.00E-04 3.00E-04 4.00E-04 5.00E-04 -0.5 Time (s) Courtesy: Lia Merminga USPAS Accelerator Physics June 2013