FLS 2012: Deterministic Approaches

Transcription

1 FLS 212: Deterministic Approaches Johan Bengtsson Mar 5, of 3

2 Outline 1. What s Known. 2. SR LS Thermodynamics (physics limitations). 3. An IBS Limited (deterministic) Approach: TME => no Chromaticity Wall. 4. Chromatic Control: (deterministic approach). 5. Pseudo-Knobs: A Leading Order (reductionist) Approach. 6. Signal Processing Closing-the-Loop : In the Control Room (deterministic approach). 8. Model Based Control by Thin Clients (deterministic approach). Challenge: When a (brute force) numerical approach doesn t cut it, how to fix it? 2 of 3

3 What s Known [1] K.W. Robinson Radiation Effect in Circular Electron Accelerators Phys. Rev. 111 (1958). [2] M. Sommer Optimization of the Emittance of Electrons (Positrons) Storage Rings LAL/RT/83-15 (1983). [3] L. Teng Minimum Emittance Lattice for Synchrotron Radiation Storage Rings FNAL/TM-1269 (1984), ANL LS-17 (1985). [4] Y. Baconnier et al Emittance Control of the PS e ± Beams Using a Robinson Wiggler NIM A235 (1985). [5] H. Wiedemann Future Development of Synchrotron Radiation Sources at Stanford PAC87. [6] G. Brown et al Operation of PEP in a Low Emittance Mode PAC87. [7] R.P. Walker et al General Design Principles for Compact Low Emittance Synchrotron Radiation Sources PAC87. [8] H. Wiedemann An Ultra-Low Emittance Mode for PEP Using Damping Wiggler NIM A266 (1988). [9] M.G. Minty et al Emittance Reduction via Dynamic RF Frequency Shift at the SLC Damping Rings SLAC-PUB-7954 (1988). [1] G. Wüstefeld et al The Analytical Lattice Approach for the Ring Design BESSY II EPAC1988. [11] V. Litvinenko Storage Ring-Based Light Sources FLS1999. [12] M. Böge et al Commissioning of the SLS Using CORBA Based Beam Dynamics Applications PAC1. [13] P. Emma, T. Raubenheimer Systematic Approach to Storage Ring Design PRST-AB 4 (21). [14] J. Guo, T. Raubenheimer Low Emittance e - /e + Storage Ring Design Using Bending Magnets with Longitudinal Gradient EPAC2. [15] R. Nagaoka, A. Wrülich Emittance Minimization with Longitudinal Dipole Field Variation NIM 575A (27). 3 of 3

4 SR LS Thermodynamics The horizontal emittance is given by (isomagnetic lattice) ε x = τ x H x D δ, σ2 δ = τ E H x D δ, N b is the no of dipoles, J x + J z = 3, F 1. No dipole gradients => J x 1. With damping wigglers, the natural horizontal emittance scales with the radiated power ε x [nm rad] = ( E [GeV]) 2 F J x N3 b ε x ε xw ε x U U + U w , σ δw σ δ = B w U w π B U U w U i.e., on behalf of σ δ. High end Insertion Devices requires σ δ of 3

5 Intrabeam Scattering (IBS) Equilibrium SR IBS ε x ε x + εx τ x ( E SR SR IBS = = ) H x ( D δ ( ρ) + Dδ ), σ δ 2 τ δ ( E SR SR IBS = )( D δ ( ρ) + Dδ ) where δ E E , H x η Tη, E η η x, η A 1 η, A 1 1 β = x η x α x β x β x. 5 of 3

6 Touschek Life Time Trade-Offs (NSLS-II CDR, 26) Lifetime, hrs for.3 nm rad Lifetime, hrs for.5 nm rad Lifetime, hrs for 1.5 nm rad Lifetime, hrs Energy acceptance, % 6 of 3

7 An IBS Limited (deterministic) Approach 1. Hor. emittance (natural): damping diffusion. 2. Optimize (globally, for Insertion Devices): Hor. Emittance 1/Hor. Chromaticity Cost of Footprint Circumference (C) Since ε x, P ρ 2 ρ bend radius. Fundamental limit is IBS: ( 2 3) ε x IBS. Cost of Operations Power Consumption (P) PETRA-3, NSLS-II, and MAX-IV avoid the chromaticity wall with damping wigglers and a 7-BA. A TME (reductionist) artifact; by ignoring (linear) chromaticity. 7 of 3

8 NSLS-II CDR (26): Parametric Evaluation =.2.25 nm rad. ε x IBS C: ~$1 M/m. 8 of 3

9 Implementations : DBA, TBA -> 7BA. Reduced peak dispersion => a stiffer (nonlinear) system (of ODEs) for chromatic control. N b 3 MAX-IV (7BA-2 => relaxed optics, by innovative engineering ε x =.26). PEP-X baseline (TME ε x =.16) -> ultimate (2.8 MAX-IV ε x =.12). USR7 (1BA-4 in the Tevatron tunnel GeV). J x J z ε x σ δ : gradient dipoles (incl. s-dependent), Robinson wigglers, orbit (i.e., dipoles ) in the quadrupoles. Insertion Devices => σ δ 1 1. F: chromatic straights (effective hor. emittance). Symmetric lattice => dispersion at the RF cavity. : damping wigglers. Requires achromatic straights => F = 3). But also ε x σ δ provide free beamlines. Insertion Devices => σ δ 1 1. Nota Bene: While facilities based on DBAs, after converting to chromatic straights, to my knowledge, have only reported the relative improvement. ε x of 3

10 Chromatic Control: First Principles Challenge: How to control the swamp of undesirable terms generated by (linear) chromatic correction for a strongly focusing lattice? M = A 1 e D V + D K2 e A Zero the undesirable terms in V; a highly over constrained problem. The most effective approach: use symmetry (reduces all terms). For example, linear achromats (in the phase-space variables): DBA, TBA, 7BA, etc. Traditional design strategies: 1. Avoidance (weakly focusing rings with high periodicity): Introduce two chromatic families and choose the working point so that systematic resonances are avoided. 2. Anti-symmetry (FODO lattices): Introduce two chromatic families separated by horizontal- and vertical phase advance of kπ, k = odd. However, this will drive h 21 and h 21 systematically. 3. Higher order achromats (strongly focusing lattices): define a unit cell, repeat it N times, and choose the phase advance so that all the 1st and 2nd order driving terms are cancelled. However, the working point is now on an integer. 1 of 3

11 Example: 5-Cell Second Order Achromat 1. Introduce two chromatic sextupole families. 2. The first order driving terms are cancelled by e.g.: Cell ν x, y ν x 2ν x 2ν y 3ν x ν x 2ν y ν x + 2ν y 1 (8/5, 3/5)=(1.6,.6) Introduce 1 more chromatic and 5 geometric (i.e., a total of 9 families => full control of all the first order driving terms); to provide leeway for the choice of working point (SLS Tech Note 9/97). 4. The required number of sextupole (-> multipole) families, placement, etc. can be evaluated & optimized by analyzing the rank conditions for the Jacobian of the driving terms (J. Bengtsson et al NIM 44, 1998). 11 of 3

12 First Order Chromatic Effects Cancelled Over 5 Cells h 21 h 21 h Im(h) Im(h) Im(h) Re(h) Re(h) Re(h) h 21 h 21 h of 3

13 First Order Geometric Effects Cancelled over 5 Cells h 111 h 21 h Im(h) Im(h) Im(h) Re(h) Re(h) Re(h) h 111 h 21 h 3 h 12 h Im(h) Im(h) Re(h) Re(h) h 12 h of 3

14 NSLS-II: Higher Order Achromats First O rde r Geometric Second Order Th i rd O rder Chromatic Geometric Geometric Cell ν x ν y ν x 3ν x ν x -2ν y ν x +2ν y 2ν x 2ν y 4ν x 4ν y 2ν x -2ν y 2ν x +2ν y 5ν x ν x -4ν y ν x +4ν y 3ν x -2ν y 3ν x +2ν y /4, 5/ /5, 6/ /6, 7/ of 3

15 Pseudo-Knobs: Leading Order (reductionist) Approach As an attempt to introduce more knobs, one may (artificially) reduce the symmetry of a multipole family. However, while a free parameter is obtained to control the leading order terms, the approach will (systematically) drive the next order(s). So, for a systematic approach (of any scheme), effects (at least) one order beyond the knobs must be included in the analysis. The impact on NSLS-II is summarized in Tech Note 9, 29: 15 of 3

16 Closing-the-Loop R + Σ u(t) h(t) y(t) - K Strategies (an iterative process): Design ( feed-forward ): model, guidelines, engineering, reality checks, etc. In the control room ( feed-back, e.g. commissioning): Model Based Control, Orbit Response Matrix, Turn-by-Turn BPM data, etc. 16 of 3

17 Beam Transfer Function & Model Based Control xt () ht () yt () RHIC, 26 (AC dipole) LEAR, 1988 (pinger) SSC, 199 (tracking) SLS, 27- (pinger) 17 of 3

18 Discrete Fourier Transform (DFT) The Discrete Fourier Transform (DFT) is defined by where x k Typical window functions N 1 = X n e i2πkn N, k = 1,,, N 1 n = N 1 1 X n = --- x N k e i2πkn N, n = 1,,, N 1 k = Rectangular: e i2πkν rect k N --- sinc( π( n Nν )), Sine: e i2πkν sin π k N sin( π( n Nν ) ) π ( n Nν ) 2 ( 1 2) 2, Hann: e i2πkν sin 2 π k N sinc 2( n Nν ) 2 ( π( n Nν 1 )) 18 of 3

19 Numerical Analysis of Fundamental Frequency It has become fashionable to use (Laskar, 1993, NAFF) N 1 Max{ X( ν) } = Max w k x k e i2πkν k = i.e., to solve numerically for a Hann window w k sin 2 πk = , k N 1 N and component wise spectrum deconvolution by Gramm-Schmidt ortogonalization. 19 of 3

20 Frequency Domain Approach: Interpolation Formula A more direct approach is to use a two-step (nonlinear) interpolation formula for the spectrum. For example, the frequency of a peak is given by: Rectangular: Sine: Hann: ν ν ν 1 N --- n , + A n 1 A n While the resolution of the discrete spectrum is only ~1 N, it is thus improved to ~1 N α, α = 234,,, respectively; i.e., ignoring the impact of noise (=> academic). Taking the effect of noise into account gives instead δν 2 Clearly, a time-domain approach has the same (fundamental) limitation. = = = 1 N --- n , 1 + A n 1 A n 1 N --- n A n 1 A n = SNR 2 N For e.g. N = 256 with 1% or 5% noise we obtain δν 4 1, 2 1, respectively.. N of 3

21 Signal Processing 11: Windowing 21 of 3

22 First Order Sextupolar Modes (SLS 9/97) 22 of 3

23 On-Line Control of First Order Driving Terms 23 of 3

24 Example: Source Analysis A x FFT 2J x ν x A y FFT 2J y ν y NSLS-II nominal spectrum ν x, y = [ 33.12, 16.19]. With a decapole component => 3ν x 2ν y 24 of 3

25 RHIC: Model Based Control (PAC7) Qx=2/3 Correction with RHIC Model Player J.Bengtsson, Y. Luo, N. Malitsky, T. Satogata, (26) RHIC Online Model sextupoles Challenges: Only two chromatic families. However, the 12 arcs have independent power supplies. UAL Model Player More knobs vs. reduced lattice symmetry. Sextupole circuits later rewired to provide 12 independent, symmetric knobs. Nikolay Malitsky, APEX Workshop 26 Status (on chromatic control) given at CERN, of 3

26 Beam Studies SLS (27) A. Streun, of 3

27 Control of Off-Momentum Aperture (SLS) A. Streun, of 3

28 Control of Nonlinear Resonances at DIAMOND (21) ν x + 2ν y = 52 compensated. 3ν x = 82 compensated. In collaboration with R. Bartolini, I. Martin, and J. Rowland. 28 of 3

29 Model Based Control by Thin Clients Client-Server Architecture for HLA Middle Layer Servers Distributed Front-Ends Ethernet Production MMLT HLA Client Client PVAccess/CAC PVAccess/CAC PVAccess Model Server PVA CAS Diagnostics Physical Device Tracy-3 PVAccess Lattice Server SQL IRMIS PVA CAS Power Supplies Physical Device Scripting HLA Client PVAccess/CAC PVAccess Channel Finder Svr SQL RDB PVA CAS RF Physical Device Control System Studio PVAccess / CAC PVAccess/CAC PVAccess/CAC Multi- Channel Arrays Svr Serves orbit, magnets, any array of channels PVA CAS Vacuum Physical Device Save/Compar e Restore Svr SQL IRMIS PVA CAS Utilities etc.., Physical Device Completed Early Development Being Extended PVAccess/CAC Magnet Conv, Response Matrix, Dispersion, etc. PVA CAS Diag & PS LS2 Simulation (Tracy-3) In collaboration with B. Dalesio CD2, 27. Improved by G. Shen, L. Yang, and J. Choi: Tracy-4: Tracy-3 interfaced to Python and Lex/ Yacc based lattice parser. Name srv, Twiss srv, etc. 29 of 3

30 Conclusions 1. We have shown how a first principles, rather than the traditional TME (reductionist) approach, provides a systematic strategy for the design of an IBS limited synchrotron light source. In particular, the insights gained from a proper understanding the scaling laws; governed by physics. 2. And summarized on how this approach was used for the NSLS-II CDR (26). In particular, how come a DBA-3 with damping wigglers, outperformed the originally proposed TBA-24 (2 SLS). 3. Similarly, MAX-IV has avoided the TME trap (i.e., the chromaticity wall ) as well, by implementing a (realistic) 7BA (with relaxed optics); by clever engineering. 4. Which recently inspired PEP-X to re-baseline. 5. We have also shown how the control theory problem for a (nonlinear) system ODEs, can be pursued all the way to the control room. By controlling the Lie generators (i.e., the equations of motion) directly. Facilitated by a scalable (aka client/server) software architecture for model-based control. 6. Bottom line, a round beam synchrotron light source is now within the horizon. 3 of 3