Number of Blocks per Pole

Diego Arbelaez

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

Slide 2

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 distribution when 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

Slide 3

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

Slide 4

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 errorSlide 5

6

Various errors

Br strength error

Br angle error bottom block Br angle error top block

Pole height error

7

Error Normalized Flux SignaturesBr strength error

Br angle error bottom block Br angle error top block

Pole height error

0.2% Br error

25 μmerror

0.1°error

0.1°error

8

Error Normalized Integral SignaturesBr strength error

Br angle error bottom block Br angle error top block

Pole height error

0.2% Br error

25 μmerror

0.1°error

0.1°error

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

Slide 9

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

Slide 10

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

Slide 11

Two Block Sort Results

Simulation Parameters: N = 260M = 10000μ = 1.32σ = 0.01μf = 2

One block per pole distribution

Improvement due to sortingTwo blocks per pole distribution

Slide 12

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

Slide 13

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°

Slide 14

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

Slide 15

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

Slide 16