1. G. Volpini, J. Rysti CERN 15 July 2015 Round Coil Superferric Magnets (RCSM) magnetic field properties 1

2. G. Volpini, J. Rysti CERN 15 July 2015 2 The INFN-funded MAGIX project is financing the development construction and test of the prototypes of the HL-LHC corrector magnets. To broaden its scientific goals, MAGIX foresees the parallel development of a MgB 2 -based solution for one magnet. This MgB 2 design is not in the baseline, and it is focused to the development of innovative and unconventional solutions, rather than to the accomplishment of all the design specifications. This work is not part of the CERN-INFN collaboration agreement KE2291/TE/HL-LHC, even if an informal collaboration is taking place. Foreword

3. G. Volpini, J. Rysti CERN 15 July 2015 3 A multipolar magnet displays symmetries that determine general features of its magnetic field. In the case of multiple coil (“conventional”) magnets, the symmetries explain well-known properties. To this respect, this approach is nice but does not teach anything new. But when coming to Round Coil Superferric Magnets (RCSM), their symmetries allow us to predict specific properties which are fairly different from what we are used to. Magnet Symmetries

4. Let’s take, as an example, a sextupole magnet. We get easily convinced that, no matter how it is done, it is invariant by a 120 degree rotation. This simply reflects the fact it is a sextupole! But there’s much more… Note: From now on, we will use a cylindrical coordinate system whose z axis is directed along the magnet (beam) axis. Example: the sextupole OPERA and Roxie models of a Hi-Lumi corrector sextupole G. Volpini, J. Rysti CERN 15 July 2015

5. “Conventional” Magnets G. Volpini, J. Rysti CERN 15 July 2015

6. These symmetries can be written as follows think of C etc. as operators: although probably correct, some issues must be checked. C P z=0 G. Volpini, J. Rysti CERN 15 July 2015 “Conventional” Magnets

7. Let’s reinvent the wheel… G. Volpini, J. Rysti CERN 15 July 2015

8. so what ???

9. G. Volpini, J. Rysti CERN 18 June 2015 Round Coil Superferric Magnet 9 Simple circular coil shape. Cost effective. We consider a sextupole configuration. Computations cross-checked between Opera and COMSOL. Kashikin, IEEE Trans. App. Sup. 20, 196 (2010).

10. 10

11. G. Volpini, J. Rysti CERN 15 July 2015 11

12. A (symmetry) rule changer G. Volpini, J. Rysti CERN 15 July 2015 -It is invariant by 120 degree rotation. -A rotation by 60 °, amounts to a “mirroring” w.r.t. a plane normal at z-axis, at z=0. No change in current. -No overall mirror symmetry. So it’s definitely different from a “conventional” magnet. The arguments in the earlier slide do not apply any more, and they are replaced by… Now, let’s have a look to the case of a RCSM with one coil. Let it be called RCSM1. Again, a sextupole can help to visualize the situation.

13. G. Volpini, J. Rysti CERN 15 July 2015 RCSM1 Magnet

14. G. Volpini, J. Rysti CERN 15 July 2015 14 a6 a9 Field Quality RCSM1 Using linear iron. Normalized with A3(z). Middle of magnet: z=0.

15. G. Volpini, J. Rysti CERN 15 July 2015 15 Main Field RCSM1 Using linear iron. Middle of magnet: z=0.

16. G. Volpini, J. Rysti CERN 15 July 2015 16 Longitudinal Field RCSM1 Using linear iron. Middle of magnet: z=0.

17. G. Volpini, J. Rysti CERN 15 July 2015 17 Now, let’s consider the case with an infinite number of RCSMs stacked on top of each other with alternating orientation (RCSM∞). Closest to a 2D case for these magnets. For the radial field, same symmetries apply as for RCSM1. Longitudinal field B z cancelled by successive magnets. Infinite Stack

18. J=10 A/mm 2 a6 a9 G. Volpini, J. Rysti CERN 15 July 2015 18 a6 a9 J=150 A/mm 2 Using ROXIE iron BH-curve. Note: z=0 is now at the ”bottom” of one magnet module. J=150 A/mm 2 Field Quality RCSM∞

19. G. Volpini, J. Rysti CERN 15 July 2015 19 A9 a9 Integrated over one magnet module. Integrated Field RCSM∞ A3

20. G. Volpini, J. Rysti CERN 15 July 2015 20 Integrated B z over one magnet in an infinite stack. Adjacent magnets in the stack give an equal and opposite contribution. Longitudinal Field RCSM∞

21. z Two magnets with mirror orientation, and reversed current (RCSM2). G. Volpini, J. Rysti CERN 15 July 2015 Two coils

22. G. Volpini, J. Rysti CERN 15 July 2015 22 Main Field RCSM2 Using linear iron.

23. G. Volpini, J. Rysti CERN 15 July 2015 23 a6 a9 a12 Field Quality RCSM2 Using linear iron. Normalized with A3(z).

24. G. Volpini, J. Rysti CERN 15 July 2015 24 Longitudinal Field RCSM2 Using linear iron.

25. G. Volpini, J. Rysti CERN 15 July 2015 25 Extending Iron length 9.0124 T*mm / 2 = 4.5062 T*mm4.5237 T*mm BUT: Limited by iron saturation. J=10 A/mm 2 100 mm 200 mm

26. i.e. independent of length in z-direction. Therefore the integrated field is proportional to the iron length, for a given value of the ampee-turns G. Volpini, J. Rysti CERN 15 July 2015 26 A B D Applying Ampère’s law E In the limit μ r =∞, contributions to integral come from AB and AE only. Pole Expansion Return yoke Extending Iron length C Field at the pole tips is then

27. G. Volpini, J. Rysti CERN 15 July 2015 27 Extending Iron length RCSM2, integrated on one module

28. G. Volpini, J. Rysti CERN 15 July 2015 28 10 A/mm 2 20 A/mm 2 RCSM2: Saturation

29. G. Volpini, J. Rysti CERN 15 July 2015 29 300 A/mm 2 RCSM2: Saturation 50 A/mm 2

30. G. Volpini, J. Rysti CERN 15 July 2015 30 RCSM2: A 3 component End of Iron Coil location 10 A/mm 2 20 A/mm 2 50 A/mm 2 300 A/mm 2

31. RCSM’s show several unusual properties: The RCSM1 has a net z component, which might be undesirable. Its even harmonics are in general nonzero, but their integral along z vanishes. The RCSM∞ is highly symmetric, and has no net z component; it can be considered a kind of quasi-2D case. Fancy, but it has no practical application. RCSM2 possesses reflection symmetry, but surprisingly it has no other symmetries. It has no net z-component, but it turns out to have a very high harmonic content, lacking those symmetries which “suppress” specific harmonics. Saturation may lead to counterintuitive behaviour. Conclusions G. Volpini, J. Rysti CERN 15 July 2015

32. 32 The End

33. (pseudo)vectors I’x = Ix I’y = Iy I’z = - Iz B’x = - Bx B’y = - By B’z = + Bz polar vector axial vector or pseudovector

34. G. Volpini, J. Rysti CERN 15 July 2015 34 150 A/mm 2