Number of Blocks per Pole

1. Number of Blocks per Pole Diego Arbelaez

2. Option – Number of Blocks per Pole • Required magnetic field tolerance of ~10-4 • For a single gap this can be achieved with one block per pole by correcting permanent magnet block errors with various shimming strategies • For example: can correct a block strength error with a pole position error • i.e. one error is compensated with a different type of error • Some errors can be compensated by performing block location sorting • For a variable gap undulator the different error signatures may not scale equally as a function of gap • It is desirable to have the pole energization be as uniform as reasonably possible • Pole energization uniformity can be improved by using multiple blocks per pole that have been sorted

3. One or Two Blocks per Pole Analysis Magnetic field error calculation • Investigate the effect of various type of errors including block strength and angle and pole position errors • Determine the scaling of these errors as a function of gap • Do the different errors scale differently with gap? • Will compensating one type of error with another type at one gap lead to errors at a different gap? Sorting simulations • Monte Carlo simulations are performed to determine how much improvement is obtained in the block Br distributionwhen two blocks are used • Simulated Annealing simulation is used for the sorting of the permanent magnet blocks within the undulator to determine the effect of the block population uniformity on field integral and phase errors

4. 2D Undulator Magnetic Model • Infinite permeability poles and permanent magnet blocks (charge sheets) • Poles are on constant scalar potential • Decompose into direct and indirect flux • Direct flux comes from permanent magnet blocks into the poles • Indirect flux is due to variations in the scalar potentials of the poles • Scalar potential of each pole is determined by setting the net flux through each pole to zero Capacitance Matrix relates the indirect flux between poles to their potential difference

5. Error Analysis • Permanent magnet block errors are modeled by excess charges • Poles errors are modeled by a boundary value perturbation Neglecting higher order term: ; For block error ; For pole error

6. Various errors Pole height error Br strength error Br angle error bottom block Br angle error top block

7. Error Normalized Flux Signatures Pole height error Br strength error 0.2% Br error 25 μm error Br angle error bottom block Br angle error top block 0.1° error 0.1° error

8. Error Normalized IntegralSignatures Pole height error Br strength error 25 μm error 0.2% Br error Br angle error bottom block Br angle error top block 0.1° error 0.1° error

9. Error Analysis Conclusions • Error analysis model allows a variety of errors to be studied with minor modifications to the model • There is not net steering caused by the above errors • The pole height error is more localized than the block strength errors • The Br strength error causes a net displacement • The normalized flux from the pole height error is fairly invariant with gap • The normalized flux for block errors varies significantly with gap • Due to different gap scaling of errors it is desirable to have the blocks energized as uniformly as possible

10. Permanent Magnet Sorting • Determine how the distribution of Br for a population of permanent magnet blocks is affected by • Number of blocks per pole • Uniformity of the population • Total number in the population • Monte Carlo simulations are performed to determine the distribution of average Br for the sorted blocks • Assume that the individual block population is normally distributed with mean μ and standard deviation σ • Blocks with abs(Br-μ) > fσ (where f is a prescribed factor) are rejected from the population

11. Monte Carlo Algorithm • Select a normally distributed random value with the assigned μ and σ • If abs(Br-μ) < fσ keep the value in the population otherwise proceed to the next random number • Repeat until there are N values in the population • Perform sorting • Repeat for M random populations of N values each Sorting: • Two blocks per pole • Generate two populations of blocks (upper and lower) • Multiply the upper blocks by the energization reduction factor • Sort the lower and upper block populations in descending order • Combine the lower block with the lowest Br and the upper block with the highest Br • Repeat for the next lowest and highest until all blocks have been paired

12. Two Block Sort Results One block per pole distribution Simulation Parameters: N = 260 M = 10000 μ = 1.32 σ = 0.01μ f = 2 Improvement due to sorting Two blocks per pole distribution

13. Simulated Annealing Sorting Algorithm • Simulated Annealing algorithm • Swap two blocks within the undulator • Compute the change in the objective function (dE) • If dE = 0 -> accept the change • If dE > 0 -> probability of accepting the change obeys Boltzmann distribution (i.e. P(dE) α exp(-dE/T) ) • Start at high T and reduce until a minimum is obtained • Objective function • First field integral uniformity • Second field integral uniformity • Phase errors • Only Br strength errors are assumed • One and two block populations from the Monte Carlo simulation are used as the input • Results are for a single gap

14. Simulated Annealing Sorting Results • One Block (standard deviation = 1% of Br) • RMS Second integral error = 0.10μT-m2 • RMS Phase error = 1.7° • Two Blocks after sort algorithm • RMS Second integral error = 0.01μT-m2 • RMS Phase error = 0.34°

15. Sorting Conclusions • Two block sort improves the uniformity of energization by ~ one order of magnitude • Further sorting can be performed by selecting the position of each block along the undulator to minimize trajectory and phase errors • After sorting tuning will be performed (e.g. by varying the pole positions) • Pole and block errors scale differently as a function of gap • It is desirable to have the pole energization be as uniform as possible to have proper tuning for the entire range of the undulator gap

16. Cost/Benefit of Options • Estimated costs for blocks: • HXU: $2M • SXU: $2.9M • One block vs. two: • VAC estimate: two ~20% more • Assume a total of 30-40% more with extra handling • Cost of 2 vs. 1 is estimated to be $1.5M - $2M • Two blocks per pole improves the energization uniformity by ~ one order of magnitude • One block makes tuning over gap range more difficult, increases risk