Some considerations and back-of-the- envelope computations on the multipole correctors

1. Some considerationsandback-of-the-envelopecomputationson themultipole correctors Giovanni Volpini, CERN, 7 March 2013

2. Basic magnet design • Goal: to perform a basicmagnet design, without FEM computations. • Assumptions made • 2D • iron with ∞ permeability • allorders (n) • Input data • Bore diameter ( 2 r0) [mm] • Intstrenghtatreferenceradius = 50 mm [T·m] • Pole field at aperture radius (r0) [T] • Pole azimuthwidthfraction [-] • Operating current [A] • SC wirediameter [mm] • Windingradialthickness [mm] • …some otherstuff… Giovanni Volpini, CERN, 7 March 2013

3. Magnetgeometry & field The Ampère’s law on the circuit OPRSQO gives the ampere·turnsrequired to achieve a givenB0on the iron pole tips (points P and Q). The flux in the irondepends on the extension of the pole, which in the figure goes from P to W. The integral can be solvedexactly. The minimum ironwidth (from T to U) must cope with thisfluxwithoutsaturating. Note: the 4-pole isusedas an example. Formulaehold for any n≥2 R The iron maximum radius can be foundsummingr1 + WT (coil heigth) + TU (ironwidth) A larger pole givesbetterfieldqualitybutwill saturate at W, and willincrease the overallironsize, because of the largerflux. U P T W O r1 r0 Q S Giovanni Volpini, CERN, 7 March 2013

4. Energy stored in the magnet The iron pole in the first quadrantextends from Q to R. Energy in the first quadranthasbeenintegrated in the area OPQRSO The contribution from the outerregionsis large: Ifweconsider an angle ROQ of wehave (allquadrants) B0 = |B(r0)| P Q R r1 θ1 S O ½π-2θ1 The contribution from the outerregionsis large: ifwerestrict to the circle of radius r0wehave, for anyordern r0 Giovanni Volpini, CERN, 7 March 2013

5. Energy stored in the magnet This large contribution to energy from thenregionoutside the free bore isnotsurprising, since the fieldgrows with increasingradius. An explicitform for the energyexists for anyvalue of the angle ROQ, whichgoeslike 1/n. Ifweintroduce a formfactor f(θ1) the energy can be expressedas • - in the case of a circle of radius r0, f = π, • - for the whole area shownbefore, assumingreasonablevalues for θ1f ~ 4-5 • MISSING (so far) • the energy in the coils; • an attempt to estimate the energy in the iron, assuming µ<∞ Giovanni Volpini, CERN, 7 March 2013

6. SC wire NbTiis the referencesolution; MgB2solutionisbeingconsidered in parallel for itspotential in terms of temperature margin. Contacts (notcontracts…) with Columbus Superconductors are in progress. The mainissueis the minimum bending radius, which in the productsmanufactured so far is80 mm or larger. We are planning measurements to verifywhether bending radiisuitable for ourdesigns can be reached. Giovanni Volpini, CERN, 7 March 2013

7. SC wire NbTiJc 9,000 A/mm2 @ 2T 1.9 K ( Ic = 609 A ) thiscorresponds to 2,700 A/mm2 @ 5T 4.2 K d = 0.5 mm α = 1.9 anyotherrequirement? Whatis the definition of «% of the load line?» Giovanni Volpini, CERN, 7 March 2013

8. Giovanni Volpini, CERN, 7 March 2013

9. Giovanni Volpini, CERN, 7 March 2013

10. Comparison Giovanni Volpini, CERN, 7 March 2013

11. a fewissues -Operating current: pro’s and con’s -Field qualityrequirements, especially for the 4-pole -Pole extension -Maximum voltagepermissible (up to 50 V the design is «lowvoltage», ifweexceedthisvaluewe can go, to, e.g. 300 V?) - Protection: must be confirmeditis OK in some cases Giovanni Volpini, CERN, 7 March 2013

12. the end Giovanni Volpini, CERN, 7 March 2013