1. HL-LHC IR Corrector Magnets Design & Construction Activity Giovanni Volpini on behalf of the LASA team CERN, 9 October 2014

2. Goals Homework: Assess the electromagnetic coupling between the corretor magnets and the forces acting between them. A full (2π) model has been developed since in the most general case no symmetry exists. One magnet is powered, with real iron and the second one (coupled) is described through its iron yoke, assuming linear iron. Poor behaviour due to the loose boundary conditions and the «mixture» of different problems (high field, current driven on one side, and «quasi magnetostatic» on the other), led to convergence problem and doubtful solutions. A simplified model has therefore been introduced, simplyifing the second magnet description, leaving out the iron yoke and considering only the end plates (flux return yoke). This restores the natural symmetry of the problem, ruling out non-physical solution, reducing comuptation time/increasing the accuracy, at the price of a somewhat less accurate description of the second magnet. We consider, so far, two cases: quadrupole and octupole Giovanni Volpini CERN 9 October 2014

3. Model source magnet: current yoke+bridge+FRY real iron Round-hole FRY coupled magnet: no current only FRY + bridge simulated linear iron µ r = 4000 Box for Maxwell e.m. stress tensor calculation d Giovanni Volpini CERN 9 October 2014 yoke bridge flux return yoke FRY bridge

4. cross-talk in the coupled magnet The magnetic induction in the FRY of the coupled magnet is mostly concentrated close to the bore, and is extremely small in the bridge connecting the FRY to the yoke (the latter is not modelized) Case d = 10 mm Giovanni Volpini CERN 9 October 2014

5. B in the coupled magnet as a function of the separation: octupole earth magnetic flux density Flux density in the coupled magnet FRY and bridge decreases exponentially with increasing separation between magnets. We can assume that the value in the yoke is even smaller, leading to a negligible excitation of the magnet. Nominal separation between iron yokes: 76.44 mm Cross check: Iron replaced w/ air in the second magnet Giovanni Volpini CERN 9 October 2014

6. B in the coupled magnet as a function of the separation: quadrupole Nominal separation between iron yokes: 76.44 mm Giovanni Volpini CERN 9 October 2014 FRY w/ 2 plates (std) FRY w/ 3 plates

7. Main component profile: octupole Giovanni Volpini CERN 9 October 2014 a 4 /b 4

8. Electromagnetic coupling free-standing magnet Small effect (if any) on the harmonic content and on the integrated strength The flux in the FRY of nearby magnet has an opposite sign, and therefore «cancels» that in the source magnet Giovanni Volpini CERN 9 October 2014

9. Forces between magnets 1 Integration of the Maxwell stress tensor on the surface of an air volume sourrounding the iron. In this case, we are interested in the net (external) force, so we neglected the surface on the ϱ -z planes; 2 An internal feature of COMSOL, which is based also on the Maxwell stress tensor; 3 Virtual work principle; 4 Helmholtz force density. Giovanni Volpini CERN 9 October 2014 1 was computed considering a surface in air encompassing the iron of the second magnet; Despite we do not know precisely how 2 works (COMSOL documentation explains that MST is integrated on the relevant surface, but it is unclear how this is precisely accomplished, since some components of B and H are not continuous across the iron surface), the results of 2 agree with 1 to within ±3%. 3 requires in our case knowledge of the energy with ppm (or ppb!) accuracy, which is unrealistic. Still it can be used to set an upper bound on the forces. 4 Nice formulation, but it requires the computation of the gradient of the permeability, which creates problems expecially at the iron boundaries. Not used in practice Force between iron yokes turned out to be harder to compute than expected. Following methods investigated and, when possible, exploited:

10. Forces between magnets Giovanni Volpini CERN 9 October 2014 Attractive force decreases exponentially, the higher orders the faster. F(z) = F(0) e -(z/λ) λ ≈ 33 mm (quadrupole) λ ≈ 20 mm (octupole) If ΔU is an upper bound for the stored energy variation changing the separation by Δz = z 2 - z 1, an upper bound for the attractive force is given by F(z 1 ) < ΔU/ λ ; λ < Δz F(z 1 ) Δz ΔU = 0.1 J F z (z 1 ) < 0.1 J/0.02 m = 5 N ΔU = 0.4 J F z (z 1 ) < 0.4 J/0.033 m = 12 N a 4 /b 4 a2a2

11. Construction activities Giovanni Volpini CERN 9 October 2014

12. Test cryostat for quadrupole Giovanni Volpini CERN 9 October 2014 D in = 515 mm L in = 3000 mm

13. new tool for winding & impregnation Now sent to teflon coating company. Due mid-October Giovanni Volpini CERN 9 October 2014

14. Design magnetic length cross-talk between magnets fringe field (“harmonics” at the magnet ends) forces between magnets (March 2014) Residual magnetization at I=0 and impact on the harmonics Cross check COMSOL results w/ Roxie (March 2014) Mechanical design (May 2014) November 2014 (new) conceptual design of all the magnets Construction & test Wind & impregnate a dummy coil (June 2014) Design the test cryostat (new) test coil w/ SC wire (July 2014) (new) next mould manufactured (Oct 2014) Next steps tbd

15. Conclusions Force computation on magnetized materials not a trivial subject; Forces and coupling seems negligible in the case of the octupole, the same statement applies most likely to the sextupole, and a fortiori to decapole and dodecapole; slightly-less-than-obvious for the quadrupole e.m. coupling: probably a model with a simplified description of the yoke (and not only the FRY and bridge) could help to sort this out. Giovanni Volpini CERN 9 October 2014

16. Thank you for your patience! Giovanni Volpini CERN 9 October 2014