1. SCU Measurements at LBNL
Diego Arbelaez (LBNL)
Superconducting Undulator R&D Review
Jan. 31, 2014

2. Introduction
Undulators must meet the trajectory and phase shake error requirements for the FEL
Magnetic field error sources
Random and systematic machining errors
Assembly errors
Accurate fabrication methods will be used in order to minimize the initial device errors
End and central tuning methods will be incorporated on the prototypes
Sufficiently accurate measurement and tuning methods must be available to meet the requirements for:
1st and 2nd field integral
Phase and phase shake
Keff

3. Error Sources and Analysis

4. Error Analysis for Coil and Pole Tolerances
Coil error
Produces no net kick (displacement does not grow with distance)
Produces a phase error
Pole error
Produces a net kick (displacement grows with distance)
Pole h l
Coil d w
Second Field Integral Error (Pole)
100 μm errors I1
I1 = 0.19 T-mm
I1 = T-mm
Second Field Integral Error (Coil)
δ = 0.21 T-mm2
δ = 0.94 T-mm2
* Tolerance = 50 T-mm2

5. Trajectory Error Scaling
Determine the standard deviation in the trajectory error for a random ensemble of undulator feature errors
Pole errors
Characterized by a kick error (I1)
Total trajectory error is given by the sum of kick errors (Ki) with a drift length (x-xi) (i.e ); scales with N3/2
Coil errors
Characterized by a displacement error (I2)
Total trajectory error is a simple random walk of individual displacement errors (i.e ); scales with N1/2
Pole Errors
Coil Errors
Trajectory errors scale with the undulator length to the power of 3/2

6. Scaling of Trajectory and Phase Errors for Untuned Devices
Random pole and coil errors with a given standard deviation are introduced using a Monte Carlo simulation for an undulator with length Lu = 3.3 m
Calculations performed for as-built undulator with no field tuning
RMS machining errors of < 2μm were measured in the ½-m long LBL prototype
Second field integral can be reduced to meet the requirements with end and central field correction mechanisms
Second Integral Error
LCLS-II requirement
Phase Shake
Lu = 3.3 m
LCLS-II requirement
End and central field tuning methods will be used to reduce the second integral error
quadratic increase with error size
linear increase with error size

7. Simulated Trajectory with Field Correction
Random errors generated using CMM-measured distribution of machining errors
Corrector locations and excitation (same for all locations) of correctors is applied
On average 11 correctors are needed to reduce the first and second integral errors to negligible levels over 3.3 m
The trajectory requirement is met for the entire range of operation with the only adjustment being the amplitude of the corrector current (same through all correctors)
11 correctors
Before correction
After correction
Lu = 3.3 m

8. Undulator Measurements at LBNL

9. Field Measurement Technology Approaches
Hall Probe (ANL)
Local field measurement
Need to know the location of the hall probe to high accuracy
Stretched wire or coil scan (ANL)
Obtain net first and second field integrals
Only length integrated information
Pulsed wire (LBNL)
Measure first and second field integrals
Measurements give integral values as a function of position along the length of the undulator

10. Pulsed Wire Method Description
Tensioned wire between two points
Part of the wire is in an external magnetic field
A current pulse is applied to the wire
The wire is subjected to the Lorentz force
A traveling wave moves along the wire
The displacement at a given point is measured
The displacement of the wire as a function of time is related to the spatial dependence of the magnetic field
Observation
point (z = 0)
Bx(z) I z y x
Traveling wave

11. Analytical Solution (Dispersion Free)
Solution for the wire motion at a given location as a function of time
A square current pulse with pulse width δt is assumed
ρ: wire mass per unit length
T: wire tension
c: wave speed
General solution:
Special cases:
: wire position at z = 0 as a function of time
DC current: ; I1 ct
δt 0: z

12. Dispersion
The flexural rigidity of the wire leads to dispersive behavior
Thin wires with lower flexural rigidity are less susceptible to dispersion
Dispersive behavior can be predicted using Euler Bernoulli theory for bending of thin rods
General Solution
Euler-Bernoulli Beam
Dispersive wave motion:
Undispersive wave motion:

13. Experimental Validation
Wire Positioning stages
Wire position sensors (referenced to undulator fiducials)
Wire motion detectors
Echo-7 Undulator

14. Wave Speed Measurement
Wave speed obtained by placing the motion sensor in two different locations and measuring the phase difference as a function of frequency in the two signal
Wire motion from magnet at two locations
Wave Speed
Fit to analytical expression

15. ECHO-7 First and Second Integral Measurement
First Integral
Second Integral
Before Dispersion Correction
After Dispersion Correction

16. Phase error calculation with upstream and downstream detectors
ECHO-7 Phase Error
Wire damping introduces error in the field integral measurement which must be compensated in the calculation of phase errors
Phase error calculation with upstream and downstream detectors
Comparison of the calculated phase errors for Hall Probe and PW measurements

17. In-vacuum Pulsed Wire System
SCU Test System
Cryogen-free cryostat (two cryo-coolers)
Pulsed wire attachment at each end of the cryostat
In-vacuum pulsed wire measurement
Decreased air damping overcome with passive damping at the ends and pulse cancelling with reverse current
In-vacuum Pulsed Wire System
Test Cryostat

18. Measurement Plan
Pulsed wire will be used as the main method during the R&D and commissioning phase for the field correction mechanism at LBNL
The pulsed wire method will be incorporated and used as one of the measurement methods in the ANL measurement system
Absolute Keff measurements will be performed using the ANL hall probe system