1. Some considerations and back-of-the-envelope computations on the multipole correctors Giovanni Volpini, CERN, 7 March 2013

2. Goal: to perform a basic magnet design, without FEM computations. Assumptions made -2D -iron with ∞ permeability -all orders (n) Input data -Bore diameter ( 2 r 0 ) [mm] -Int strenght at reference radius = 50 mm [T·m] -Pole field at aperture radius (r 0 ) [T] -Pole azimuth width fraction [-] -Operating current [A] -SC wire diameter [mm] -Winding radial thickness [mm] -…some other stuff… Basic magnet design Giovanni Volpini, CERN, 7 March 2013

3. r0r0 O P Q R S Magnet geometry & field r1r1 T U W The iron maximum radius can be found summing r 1 + WT (coil heigth) + TU (iron width) A larger pole gives better field quality but will saturate at W, and will increase the overall iron size, because of the larger flux. Giovanni Volpini, CERN, 7 March 2013 Note: the 4-pole is used as an example. Formulae hold for any n≥2

4. r0r0 r1r1 θ1θ1 O P Q R S The iron pole in the first quadrant extends from Q to R. Energy in the first quadrant has been integrated in the area OPQRSO ½π-2θ 1 Energy stored in the magnet The contribution from the outer regions is large: if we restrict to the circle of radius r 0 we have, for any order n Giovanni Volpini, CERN, 7 March 2013

5. This large contribution to energy from then region outside the free bore is not surprising, since the field grows with increasing radius. An explicit form for the energy exists for any value of the angle ROQ, which goes like 1/n. If we introduce a form factor f(θ 1 ) the energy can be expressed as Energy stored in the magnet - in the case of a circle of radius r 0, f = π, - for the whole area shown before, assuming reasonable values for θ 1 f ~ 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 NbTi is the reference solution; MgB 2 solution is being considered in parallel for its potential in terms of temperature margin. Contacts (not contracts…) with Columbus Superconductors are in progress. The main issue is the minimum bending radius, which in the products manufactured so far is 80 mm or larger. We are planning measurements to verify whether bending radii suitable for our designs can be reached. Giovanni Volpini, CERN, 7 March 2013

7. SC wire NbTi Jc 9,000 A/mm 2 @ 2T 1.9 K ( Ic = 609 A ) this corresponds to 2,700 A/mm 2 @ 5T 4.2 K d = 0.5 mm α = 1.9 any other requirement? What is the definition of «% of the load line?» Giovanni Volpini, CERN, 7 March 2013

10. Comparison Giovanni Volpini, CERN, 7 March 2013

11. a few issues -Operating current: pro’s and con’s -Field quality requirements, especially for the 4-pole -Pole extension -Maximum voltage permissible (up to 50 V the design is «low voltage», if we exceed this value we can go, to, e.g. 300 V?) - Protection: must be confirmed it is OK in some cases Giovanni Volpini, CERN, 7 March 2013

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