ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD IMPROVING ACCURACY AND ROBUSTNESS OF LINEAR OPTICS IMPROVING ACCURACY AND ROBUSTNESS OF LINEAR OPTICS MEASUREMENTS MEASUREMENTS PhD student at CERN and Andreas Wegscheider Hamburg University
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
MOTIVATION MOTIVATION • Measurement and correction of focusing errors is of great importance in circular accelerators - colliders as well as light sources • N-BPM method has been used in various machines - LHC - ALBA - ESRF • Want to improve existing measurement techniques. N-BPM method used Monte Carlo simulations to get systematic errors • time consuming • failed for pushed optics
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
β FUNCTION MEASUREMENT FUNCTION MEASUREMENT beta from phase , 3-BPM method , N-BPM method
ϕ j ϕ ( s ) ϕ ij ) ) − cot( cot( s i β ( ) = β ) − cot( ϕ i s i ϕ ( ) x CO Q cot( x CO s i s i x ( ) = A cos(2 πQ + ϕ ( )) + s i ) ϕ m ij ϕ m ik = − ϕ ij ϕ ik BETA FROM PHASE BETA FROM PHASE The position of the beam at a position in the ring is : machine tune, : betatron phase, : closed orbit Measured with Beam Position Monitors (BPMs). The phase is derived by a harmonic analysis of the BPM signal. from phase using three BPMs via the following formula: Assumption: relatively small lattice imperfections between the BPMs. blue : probed BPM (BPM i) red : used BPM ( j, k )
ϕ m ϕ ij ij ϕ m ) ) − cot( cot( ϕ ik ) ik ) − cot( cot( s i β ( ) = n ∈ ℕ ϕ ij ≈ nπ BETA FROM PHASE BETA FROM PHASE sensitive to the position of the BPMs with respect to each other. The cotangens enhances phase measurement errors if 10 5 cotangens 0 −5 −10 −1 −0.5 0 0.5 phase advance [TWOPI] Export to plot.ly »
SKIP BPMS / USE MULTIPLE COMBINATIONS SKIP BPMS / USE MULTIPLE COMBINATIONS • Skipping BPMs to avoid unpreferable phase advances. But lattice imperfections deteriorate the quality of the result • Take into account possible sources of errors between the BPMs • Use more than one combination to increase the amount of information N-BPM method
( i , , β ) ORIGINAL N-BPM METHOD ORIGINAL N-BPM METHOD To get the function at the position of a BPM i: calculate several combinations and j l k l use the mean value .
ORIGINAL N-BPM METHOD ORIGINAL N-BPM METHOD But there are errors and far-away BPMs yield worse data.
ij ) − cot( ik g l = g l l ∑ ( ) β = ik l ϕ m ϕ m ) ij l cot( ϕ ik l ϕ ij l β ) ) − cot( ( i , , cot( ) ( ) = ORIGINAL N-BPM METHOD ORIGINAL N-BPM METHOD To get the function at the position of a BPM i: calculate several combinations j l k l β l s i The best estimation for the beta function is calculates by β l s i g l The weights are determined by a least-squares estimation ∑ k V −1 ∑ i , j V −1
δϵ =0 Δ ϕ , Δ ⎟ ⎟ ⎟ M = diag ( , … , ) ϵ 1 ϵ M ϵ λ , Δ s , … β n K 1 V = TMT −1 T = T lλ ∂ β l ∂ ϵ λ ∣ ∣ ∣ ⎠ ⎞ ⋮ ij β 1 ⎜ ⎜ ⎜ ⎝ ⎛ β ⃗ = β ⃗ V = Cov [ ] = g l ik ORIGINAL N-BPM METHOD ORIGINAL N-BPM METHOD ∑ k V −1 ∑ i , j V −1 is the covariance matrix of It can be calculated from the diagonal error matrix where the are all the sources of error ( ): where is the Jacobian
ORIGINAL N-BPM METHOD ORIGINAL N-BPM METHOD • Only statistical errors were calculated analytically. Systematic errors were calculated by Monte Carlo simulations • Yields only approximated results and is time consuming . (LHC has many different optics) • Pushed optics can cause a crash of the simulations.
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD analytical calculation of the covariance matrix , removal of bad BPM combinations
{ δs } acc , { δs ∂ K μ lμ T s ( ) ∂ β l ∂ s ν lν ( ) = β ( ϕ , { δ , …) = ( + Δ β ( ϕ , { δ , { δs , …) K 1 } acc } acc β m K 1 } acc } acc { δK 1 } acc ∂ β l ) T K = T ) T ϕ lλ T s = T K = ∂ ϕ λ T ϕ ( ) ∂ β l CALCULATION OF THE CORRELATION MATRIX CALCULATION OF THE CORRELATION MATRIX The Jacobian can be split into blocks T ϕ T K T s T = ( : phase uncertainties, : magnet field uncertainties, : longitudinal misalignments The three parts of the matrix can be calculated: The phase part is already known, for the others we need a relationship β real s i : quadrupolar field errors in the accelerator : longitudinal misalignments (BPMs and quads).
ij − ij l ϕ m g ij l g ik l s i g ij = sgn( i − j ) g ij δ δ − + δ 1 ( ) s j 1 ( ) s i wj ϕ m ik l + cot β β − cot [ ( ) − 2 ( ) ≈ − cot cot ϕ ij l ϕ ik l EFFECT OF IMPERFECTIONS ON THE EFFECT OF IMPERFECTIONS ON THE FUNCTION FUNCTION AND ITS MEASUREMENT AND ITS MEASUREMENT new formula to calculate function from • phase advances • quadrupolar field errors • longitudinal BPM misalignments • longitudinal quadrupole misalignments β m s i α m s i β l s i ( ) δ ] where the terms collect the dependency on lattice imperfections . w K w ,1 sin 2 ϕ m ∑ w ∈ I β m β m s j β m s i sin 2 ϕ m
ik l sgn( i − ) ( − ) sgn( i − ) j l ijl ( ) ( ) λ δ i λ k l i ikl ( ) ( ) λ δ i λ cot − cot ϕ m ij l ϕ m ( − lλ ik l = ∓ ( λ ) − ( λ ) ) lλ ( ) ( ) cot − cot ϕ m ij l ϕ m A ij l A ik l ± ( ) = −2 CALCULATION OF THE CORRELATION MATRIX - CALCULATION OF THE CORRELATION MATRIX - CONCLUSION CONCLUSION sin 2 ϕ λj l sin 2 ϕ λk l β m s i β m s λ T K × ( sin 2 ϕ ij l sin 2 ϕ ik l and: β m s i β m s i β m s jl δ j l ) − β m s kl δ k l sin 2 ϕ m sin 2 ϕ m α m s i δ λ T s
δ = 2 π × 10 −2 , ik l ϕ m ij l ϕ m ) ) − cot( cot( ϕ ik l ϕ ij l ) ) − cot( cot( ( ) = ik l , , n ∈ ℕ Δ ϕ ∈ [ nπ − δ , nπ + δ ] δ REMOVAL OF BAD BPM COMBINATIONS REMOVAL OF BAD BPM COMBINATIONS filter BPM combinations by phase advances bad : for , threshold . • enhances phase errors • numerically unstable If any of the four phase advances in ϕ ij l ϕ ik l ϕ m ij l ϕ m β l s i is bad, the corresponding BPM combination is disregarded . 2016 and 2017 for the LHC: .
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
SIMULATIONS SIMULATIONS Evaluation of the new method
σx σK TEST SETUP TEST SETUP • generate many test lattices with randomly distributed errors • simulate measurement by tracking a single particle via PTC • use the LHC-OMC tools to analyse the data Standard deviation of introduced errors: [mm] [mm] K 10 −4 σs MQ 18 1.0 - MQM 12 1.0 - MQY 11 1.0 - MQX 4 1.0 - MQW 15 1.0 - MQT 75 1.0 - MS - - 0.3 BPM - 1.0 -
β ∗ NOMINAL LATTICE NOMINAL LATTICE 40 cm | collision tunes | collision energy
β ∗ HL-LHC LATTICE HL-LHC LATTICE 10 cm | collision tunes | collision energy | ATS optics
ACCURACY AND PRECISION OF THE METHODS ACCURACY AND PRECISION OF THE METHODS Nominal lattice HL-LHC
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
MEASUREMENTS MEASUREMENTS ATS machine development , 10cm β ∗
= 10 cm β ∗ ATS MD IN OCTOBER 2016 ATS MD IN OCTOBER 2016 Beta beating plot at 1000 3-BPM N-BPM 500 beta beating [%] 0 −500 −1000 0 5k 10k 15k 20k 25k s [m] Export to plot.ly »
MOTIVATION MOTIVATION BETA FUNCTION MEASUREMENT BETA FUNCTION MEASUREMENT ANALYTICAL N-BPM METHOD ANALYTICAL N-BPM METHOD SIMULATIONS SIMULATIONS MEASUREMENTS MEASUREMENTS CONCLUSIONS CONCLUSIONS
CONCLUSIONS CONCLUSIONS new method has been developed fully analytical calculation of the covariance matrix • faster • more accurate • avoids complication from failing simulations successfully used at CERN : • LHC measurements and optics correction (commisioning, machine development) • first measurements at PS • first measurements at PS Booster method independent of accelerator and accelerator type • Analytical N-BPM method can be used with any kind of accelerator • linear optics analysis toolbox ready for integration of new accelerators
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