A time domain modeling and simulation tool for beam-cavity interactions in LER and HER rings at PEP II are presented. The motivation for this tool is to explore the stability margins and performance limits of PEP II RF systems at higher currents and upgraded RF configurations. It also serves as test bed for new control algorithms and can define the ultimate limits of the architecture. The time domain program captures the dynamical behavior of the beam-cavity interaction based on a reduced model. The ring current is represented by macro-bunches. Multiple RF station in the ring are represented via one or two macro-cavities. Each macro-cavity captures the overall behavior of all the 2 or 4 cavity RF station. Station models include nonlinear elements in the klystron and signal processing. This allows modeling the principal longitudinal impedance control loops interacting with the longitudinal beam model. Validation of simulation tool is in progress by comparing the measured growth rates for both LER and HER rings with simulation results. The simulated behavior of both machines at high currents are presented comparing different control strategies and the effect of non-linear klystrons in the growth rates

Fox, J. D.

Mastorides, T.

Rivetta, Claudio

Teytelman, D.

Van Winkle, D.

UNT Digital Library

SLAC-PUB-12375Modeling and simulation of longitudinal dynamics for LER-HERPEP II Rings1C. Rivetta, T. Mastorides, J. D. Fox, D. Teytelman, D. Van WinkleStanford Linear Accelerator Center, Menlo Park, CA 94025, USAAbstractA time domain modeling and simulation tool for beam-cavity interactions in LER and HERrings at PEP II are presented. The motivation for this tool is to explore the stability margins andperformance limits of PEP II RF systems at higher currents and upgraded RF configurations. It alsoserves as test bed for new control algorithms and can define the ultimate limits of the architecture.The time domain program captures the dynamical behavior of the beam-cavity interaction basedon a reduced model. The ring current is represented by macro-bunches. Multiple RF station in thering are represented via one or two macro-cavities. Each macro-cavity captures the overall behavio ofall the 2 or 4 cavity RF station. Station models include nonlinear elements in the klystron and signalprocessing. This allows modeling the principal longitudinal impedance control loops interacting withthe longitudinal beam model.Validation of simulation tool is in progress by comparing the measured growth rates for both LERand HER rings with simulation results. The simulated behavior of both machines at high currents arepresented comparing different control strategies and the effect of non-linear klystrons in the growthrates.Presented at the European Accelerator ConferenceEdinburgh, UK, June 26-30 20061Work supported by U.S. Department of Energy contract DE-AC02-76SF00515.February 2007MODELING AND SIMULATION OF LONGITUDINAL DYNAMICS FORLER-HER PEP II RINGS∗C. Rivetta, T. Mastorides, J.D. Fox, D. Teytelman, D. Van WinkleStanford Linear Accelerator CenterStanford, CA 94309, USAAbstractA time domain dynamic modeling and simulation toolfor beam-cavity interactions in LER and HER rings at PEPII are presented. The motivation for this tool is to explorethe stability margins and performance limits of PEP II RFsystems at higher currents and upgraded RF configurations.It also serves as test bed for new control algorithms and candefine the ultimate limits of the architecture.The time domain program captures the dynamical be-havior of the beam-cavity-LLRF interaction based on a re-duced model. The ring current is represented by macro-bunches. Multiple RF stations in the ring are representedvia one or two macro-cavities. Each macro-cavity capturesthe overall behavior of all the 2 or 4 cavity RF stations.Station models include nonlinear elements in the klystronand signal processing. This allows modeling the principallongitudinal impedance control loops interacting with thelongitudinal beam model.Validation of simulation tool is in progress by compar-ing the measured growth rates for both LER and HERrings with simulation results. The simulated behavior ofLER at high currents is presented comparing different con-trol strategies and the effect of non-linear klystrons in thegrowth rates.SYSTEM AND MODEL DESCRIPTIONThe PEP-II RF stations comprise 1.2 MW 476 MHzklystrons with either 2 or 4 normal-conducting RF cavitieswith HOM dampers [1]. The LLRF systems include comb(a second order IIR filter) and direct loop feedback pathsto reduce cavity impedances seen by the beam. Despitethe LLRF feedback, the beam exhibits low mode coupled-bunch instabilities at operating currents due to the funda-mental impedance, and a special ”woofer” feedback chan-nel is required to control low mode instabilities [2]. Thestations also incorporate numerous slow regulating loopswhich control cavity tuners, HV power supply voltage,compensate for gap transient effects, etc. [1]. The sys-tem described is depicted in the simplified block diagramin Fig. 1.The simulation is focused on understanding the dynamicinteraction among the beam, the cavities and the fast LLRFfeedback loops. This tool is developed as a block diagramin Simulink, which uses the system parameters calculatedin Matlab [3]. The simulation is an update of a previous∗Work supported by the U.S. Department of Energy under contract #DE-AC02-76SF00515CavityComb LoopDirect LoopCable DelayBeamFeedforwardReferenceLongitudinalFeedbackΣΟΣΙ KlystronDriver +Figure 1: Simplified System Block Diagram. TransferFunction measured between I(nput) & O(utput).work developed by Rich Tighe [4]. To optimize computa-tion time and complexity only elements in Fig. 1 with dy-namics in the order of the low-order mode beam dynamicsare included. Slow feedback and control systems are con-sidered as constant, defining the operation point of the sys-tem in the time frame where the simulation is performed.The particle beam is represented via a variable number ofmacrobunches comparable to the comb samples per turn,rather than the 1746 physical bunches, which fully resolvesthe low frequency beam modes (e.g. modes: -18 to 17, forthe 36 macrobunch case) and interactions with the RF fun-damental impedance.Growth rates depend on the real part of the effectiveimpedance presented to the beam by all the RF cavities in-cluded in the ring [5]. The growth rate corresponding to thelth characteristic beam mode is given byσl = R(Λl) = −dr + παef2rfI0E0hωsR(Z‖eﬀ(lω0 +ωs)), (1)whereR(Z‖eﬀ(ω)) represents the effective impedance pre-sented by all the stations to the beam. Detailed modelsof klystrons, including non-linearities and the frequencyresponse, were considered to analyze both the limits ingrowth rate reduction due to the feedback system and dis-crepancies between stations. The simulation can be used topredict stability in future operation points, as well as studythe effectiveness of possible additions and modifications tothe RF stations.The simulation models the RF signals in baseband anduses the in-phase/quadrature formalism to represent them.The fast feedback loops at each RF station modify the cav-ity impedance asZ‖eﬀi (ω) = (I + G(ω)H(ω))−1Zsti(ω), (2)where G(ω)H(ω) corresponds to the return ratio of the sta-tion. Parameters in the control loops and the stations definethe frequency response of the system.RF CAVITY IMPEDANCE CONTROLEq. 1 expresses the particle beam stability whereas theterm in parentheses of Eq. 2 corresponds to the stability ofthe RF feedback loops. The LLRF station feedback param-eters (direct and comb loop gain and phase) are optimizedto minimize the overall station impedance Z ‖eﬀi (lω0 + ωs)at frequencies lω0 + ωs, corresponding to the beam pertur-bation, compatible with stability performance criteria forthe RF loops (e.g. gain/phase margins). In PEP-II op-erations they are calculated using a non invasive methodthat starts with the identification of the transfer function ofeach station [6]. A similar method is used in the simulationto adjust the parameters of the macrostation. To achievegood agreement between the simulation and the physicalsystem, it is important to define in the simulation an ef-fective impedance interacting with the beam equal to thereal impedance presented by the RF stations to the beam.From (1) and (2), it is important to observe that this is possi-ble only if there is agreement between the transfer functionmeasured per station and the transfer function and returnratio defined in the simulation.Since the transfer function relationship tracks the growthrate consistency, it is reasonable to use the transfer func-tions for verifying the simulation model. This was done fordifferent operating points, but the analysis below is in LERat 1400 mA. The transfer function is measured by playing anoise file in the input and reading the response at the outputas marked in Fig. 1. The time domain simulation includesklystron non-linearities and frequency response. In Fig. 2and Fig. 3 the collected data are shown in red for measuredand simulated transfer functions respectively. These are fit-ted to the same 5 parameter linear system model [6] shownin green. The value of the model is to compare the resulting−1000 −800 −600 −400 −200 0 200 400 600 800 1000−20−10010Gain (dB)Direct: Fr = 475.9±0.1 MHz; G = 5.268±0.009; Td = 430.8±0.6 ns; φ = 164.1±0.1 deg  FitData−1000 −800 −600 −400 −200 0 200 400 600 800 1000−2000200Frequency (kHz)Phase (degrees)Comb: Gc = 0.2042±0.0009; Tc = 5590±2 ns; φc = 30.1±0.3 degFigure 2: Transfer Function and parameters of operatingstation in LER at 1400 mA.sets of parameters extracted from the physical system andsimulation; thus, providing evidence of convergence. Fromthese figures we can clearly see the agreement between thedata and fit, which shows the accuracy of the fitting tools.The measured and simulated transfer functions as well asthe fitted parameters are very close, providing confidence−1000 −800 −600 −400 −200 0 200 400 600 800 1000−20−100Gain (dB)Direct: Fr = 475.851±0.002 MHz; G = 5.22±0.03; Td = 453±2 ns; φ = 164.5±0.3 deg  FitData−1000 −800 −600 −400 −200 0 200 400 600 800 1000−200−1000100200Frequency (kHz)Phase (degrees)Comb: Gc = 0.204±0.002; Tc = 5589±7 ns; φc = 31±1 degFigure 3: Transfer Function and parameters of macrosta-tion from non-linear simulation in LER at 1400 mA.that the growth rates will also be comparable. The smalldiscrepancies can be attributed it to parasitic coupling be-tween the in-phase and quadrature components to be ana-lyzed in a subsequent publication.It is important to note that the klystron frequency re-sponses are different among physical stations. One resultof the simulation was to show that the growth rates are verysensitive to these variations. Thus, either the growth rateshave to be computed for each station and averaged or amacroklystron has to be developed that will represent thewhole ring. The later case is presented.GROWTH RATE MEASUREMENTSThe essential measurements from the simulation are themodal growth rates since these are used to determine beamstability. The system is perturbed from equilibrium andthe beam modes evolve in time. In PEP-II operations themodal growth rates are estimated via grow/damp experi-ments opening the longitudinal feedback loop [7]. Sim-ilarly in the simulation, we let the beam modes evolveand study the interaction between the RF station and thebeam [2]. To achieve consistency between the physicalsystem and the simulation, the same growth rate extrac-tion tools are used to determine the growth rates in eachcase. For the simulation case, the growth rates are thencompared to the expected damping rates from the longi-tudinal feedback loop, providing a quantitative measureof stability margins for each mode, in contrast to earlierwork [4]. The natural complex frequency, whose real partcorresponds to the growth rate and the imaginary to the os-cillation frequency, is fitted to the evolving modes (shownon the left in Fig. 4). Thus, we differentiate between sta-ble/unstable modes and can select the beam mode with thehighest growth rate. The growth rates for modes -10 to 10and their oscillation frequencies can be seen on the right inFig. 4 for LER at 2500 mA. Modes -3 to -5 are usually themost unstable modes due to cavity detuning with increas-ing beam current. Fig. 5 compares the growth rates fromthe physical system and the simulation for both the macro-station and the individual station (due to different klystronresponses) at 1400 and 2500 mA. The simulation not onlyreproduces the form of the most unstable growth rates forvarious beam currents, but it also agrees with the physi-cal system in the number of that most unstable mode. The00.511.5−1001000.010.020.03time (ms)LER ring − modal analysismode numberdegrees−10 −5 0 5 10−202Growth ratesms−1−10 −5 0 5 10400042004400Oscillation frequenciesMode numberFrequency (Hz)Figure 4: Low Frequency Beam Modes (LER at 2500 mA).discrepancy at low currents will be investigated conductingdedicated measurements on the physical system.1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.400.511.522.533.54Beam Current (A)ms−1  MeasuredMacroklystronPredictionsIndividual KlystronsFigure 5: Measured and Simulated Growth Rates (Simu-lated for each station as well as the macroklystron). Ex-trapolated GRs are dashed. Most unstable mode is −3.APPLICATIONS AND PREDICTIONSThe value of this simulation tool is the ability to studythe effect of different parameters in the stability and perfor-mance of the system without requiring time from the realmachine. Additionally, simulations allow analysis of dif-ferent system configurations and parameter combinationsthat are not directly applicable to the physical machinewithout major changes. These studies have helped to un-derstand the sensitivity of the growth rates on certain pa-rameters included in the control loops.An example is the study of the impact of the comb filterphase rotation on the growth rates. The original criteriumused to configure the direct and comb loops was to maxi-mize the stability margin (gain and phase margins) of theRF feedback loop. The studies showed that this criteriumcomes with a tradeoff to the growth rates. We now under-stand that achieving great improvement in the growth rateswith a relatively small reduction of the stability margin ispossible. Details of this study are summarized in Fig. 6,where the growth rate of the dominant unstable mode isplotted versus the comb filter phase rotation. This plotcombine simulation results with the average growth ratemeasured from the LER ring operating at 1400 mA. Thecomb filter phase of 0o is defined by the maximum stabil-ity margin criterium. These results have already been ap-plied allowing increased beam stability margin. The con-sistency of the error margin between measured and esti-mated growth rates give a certain confidence to predict thegrowth rates of the machine at higher currents. Growthrates at higher currents are calculated by simulation anddepicted in Fig. 5. The dashed lines show the predictionat beam currents larger than 2500 mA in the LER ring.−10 −5 0 5 10 15 20 25 30 3500.10.20.30.40.5Comb Filter Phase (deg)Growth Rate (ms−1)  MeasuredSimulatedFigure 6: Measured and simulated growth rates vs. combphase rotation. Agreement in both the general form andmost unstable mode number.These numbers can be compared to the expected dampingrates for the corresponding operation point [2] and providea sense for the stability of the beam at those higher currents.CONCLUSIONSThe simulation of the PEP-II rings is a close representa-tion of the actual system. As such, it can predict the perfor-mance limits of the LLRF systems at higher currents andstudy the effectiveness of upgrades or their optimal config-urations. It also provides insight of the system and sug-gestions for optimal tuning, as with the comb rotation, theanalysis of the effect of the variations in the klystron re-sponses as well as the ability to separate the stability of theparticle beam from that of the LLRF. There are many fur-ther applications after the presented verification which willbe studied in the future.One of the important features of this tool is the adapt-ability to simulate the interaction between the RF stationsand the beam for other systems. The interaction betweenthe different parts of the algorithm and Simulink is througha parameter structure, which can be easily modified. Thus,the simulation was easily adapted to be used for modelingSPEAR to study Robinson instability and it was also mod-ified to help with the design of the Klystron Linearizer [8].REFERENCES[1] P. Corredoura. “Architecture and Performance of the PEP-IILow-Level RF System”, SLAC-PUB-8124, March 1999.[2] D. Teytelman et. al., “Operating Performance of the LowGroup Delay Woofer Channel in PEP-II”, SLAC-PUB-11253, May 2005.[3] Simulink, The Mathworks, Natick, MA 01760 USA.[4] R. Tighe, “RF Feedback Simulation Results for PEP-II”,SLAC-PUB-950-6856, June 1995.[5] F. Pedersen, “A Novel RF Cavity Tuning Feedback Schemefor Heavy Beam Loading”, IEEE Trans. Nucl. Sci. NS-32,p.2138, 1985.[6] D.Teytelman,“A Non-invasive Technique Low Level RFFeedback Loops in PEP-II”, SLAC-PUB-11252, May 2005.[7] J. Fox et. al., “Multi-Bunch Instability Diagnostics ViaDigital Feedback Systems at PEP-II, DAΦNE, ALS andSPEAR”, SLAC-PUB-8128, June 1998.[8] J. Fox et. al.,“Klystron Linearizer For Use With 1.2 MW 476Mhz Klystrons in PEP-II RF Systems”, SLAC-PUB-11228,May 2005.

Modeling and Simulation of Longitudinal Dynamics for LER-HER PEP II Rings

https://digital.library.unt.edu/ark:/67531/metadc890484/m2/1/high_res_d/900593.pdf

Modeling and Simulation of Longitudinal Dynamics for LER-HER PEP II Rings

Abstract

Similar works

Full text

Available Versions

UNT Digital Library