Cable and Magnet Losses Luca.Bottura@cern.ch THERMOMAG LPNHE – Université Pierre & Marie Curie 19.-20.11.2007

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

Loss targets for fast-ramped SC magnets at CERN PS SPS Present electrical consumption (MW) 8 55 Allocated power for cryogenics(1) 2 14 Cold power(2) (kW) 6.7 46 Total magnet length(3) (m) 810 6200 Maximum allowable thermal load (W/m) 7.5 Notes: allow 1/4 of total power consumption for cryoplant, leaving 3/4 to other systems at 4.2 K, assuming 300 WRT/Wcold for main lattice magnets, using present optics for PS2, and an estimate for SPS+ The magnet loss must be limited to ≤ 5 W/m

A prototypical magnet design in the 3…15 T range Nominal dipole field [T] 4.5 Ramp up/down time [s] 3 Field ramp-rate [T/s] 1.5 Cycle time 12 Coil inner diameter [mm] 100 Nominal current [A] 3200 Operating temperature [K] Length [m] 6 Mass [tons] 7.6 Sample SPS+ magnet design by courtesy of G. Kirby, CERN AC loss calculation by A. Verweij, CERN

Volumetric power density A 4.5 T magnet requires ≈ 2 dm3/m of SC coil A load of ≈ 5 W/m of magnet corresponds to a power density of ≈ 2500 W/m3 of coil Dcoil = 150 mm Dcoil = 100 mm Bmax = 4.5 T Bmax = 2.8 T Vcoil ≈ Bmax1.3 Vcoil ≈ Dcoil SPS+ PS2b

This corresponds to ≈ 0.5 % of the delivered power density Re-cap Although each magnet is different… … ball-park target heat loads are 5 to 10 W/m of magnet … this corresponds to relatively small volumetric energy input (2.5 to 5 kW/m3), much below the stability limit … the major concern is hence heat removal from the coil to the proximity cryogenics What are the internal loads ? How difficult is it to reach this target ? This corresponds to ≈ 0.5 % of the delivered power density '''= 1/0 Bmax * (dB/dt)max

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

Internal loss mechanisms in superconducting magnets AC field/current AC loss in strands and cables Hysteresis in filaments Coupling in strands and cables Dynamic resistance Eddy currents in metallic parts Hysteresis loss in the ferromagnetic yoke Joule heating (joints, index heating)

Hysteresis in SC filaments Refs: Wilson, and the RHEL Group loss per unit strand volume for  < 1 for  > 1 B Bmax dfil

Hysteresis loss for physicists qhB3 qhB

Hysteresis loss for engineers Small filaments are better at Bmax > Bp Effect of Jc(B) makes qh sub-linear at large Bmax qhB qhB3 Large filaments are better at Bmax < Bp

Difficult small filaments - 1/2 S. Le Naour, et al. Filament distortion Magnetization, and AC loss is proportional to Jc dfil. For the same area, distorted filaments always have a bigger loss than round Proximity Coupling When the thickness of matrix between the filaments gets ≤ 1/3 μm Cooper pairs can tunnel across the matrix, thereby increasing the effective filament size. Distorted filaments are more likely to be coupled. Strongest at low fields (coupling Jc has a low critical field Bc) A. Ghosh, et al. 0M (mT) Large coupling at dfil < 3 m for Cu matrix dfil (m)

Difficult small filaments - 2/2 Jc Small filaments are associated with indication of reduced Jc Cost It's a lot of filaments: dfil = 3μm D = 0.85mm Cu:NbTi = 1.6 single stacking Dbillet = 300mm N = 31000 ! Jc > 2500 A/mm2 dfil < 3 m

Re-cap on hysteresis Small filaments are required for loss reduction in magnets cycled at coil fields larger than ≈ 1 T A good NbTi strand (Jc(5T, 4.2K) > 2500 A/mm2) with achievable filament diameter (dfil ≈ 3 m) has a loss per unit (superconductor) volume (unipolar cycle 0-Bmax-0): At Bmax = 4.5 T we have E = 47 mJ/cm3 of NbTi This leads in our prototypical magnet to an AC loss of 2 W/m (average over a 12 s cycle), i.e. 40 % of the allocated thermal load budget J0 = 2.7 1010 A/m2; B0 = 0.65 T

Coupling in strands and cables filaments or strands periodic excitation with an amplitude B (peak to peak) Short time constant  is better in low frequency regime  << 1 Long time constant  is better in high frequency regime  >> 1 dB/dt response to impulse I t t

Strands coupling loss at low frequency Refs: Wilson, Carr, Turck,… B D Bmax m t D Dfil

Time constants Cu-Ni Cu-Mn Resistivity Twist pitch Barriers (Nb, Ni) or alloys (CuNi, CuMn, CuSi, or CuSn) are used to increase the transverse resistivity Attention should be paid not to affect the longitudinal resistivity (required for stability and protection) Twist pitch Rule-of-thumb Lp ≥ 10 Dstrand At Lp ≈ 5 Dstrand filaments start breaking and Jc drops Cu-Mn Collection by J. Kaugerts, ECOMAG-05

Re-cap on strand coupling Resistive barriers (Nb, CuNi ) and matrices (CuNi, CuMn, CuSi) are required for loss reduction in magnets cycled at ramp-rates faster than ≈ 1 T/s A strand with achievable time constant ( ≈ 0.5 m/s) has a loss per unit (strand) volume (unipolar cycle 0-Bmax-0): At Bmax = 4.5 T and dB/dt = 1.5 T/s we have E = 10 mJ/cm3 of strand This leads in our prototypical magnet to an AC loss of 0.5 W/m (average over a 12 s cycle), i.e. 10 % of the allocated thermal load budget

Cable coupling loss at low frequency SC Rutherford cable in transverse field eddy current loops  dB/dt cross-over contact Rc adjacent contact Ra w t aspect ratio =w/t >> 1

Rc vs. Ra, face-on vs. edge-on Refs: Morgan, Sytnikov, Wilson, Collings and Sumption, Verweij B B// w t The loss caused by Rc is much larger than that associated with Ra (typically a factor 50 in a cable with 30 strands and comparable resistances) The loss caused by “face-on” field changes is much larger than that associated with “edge-on” field changes (typically a factor 200 in a cable with 30 strands, comparable resistances and the same field changes)

Control over Rc and Ra Rc dominates Cable Edge effects Surface coating (SnAg, Ni, CuxO, Al2O3, CrxOy, …) modify both Rc (desirable) and Ra (not necessary, low values best for stability) in the range of few  to few 100’s  A foil (SS, CuZn, CuNi, …) in the cable increases Rc in the range of few m to few 10’s m, without affecting Ra Cable production, storage and operation can have a large impact on the final value of Rc and Ra Cable Edge effects Rc and Ra can be largely affected by strand deformation at cross-overs at cabling edges and cause loss concentration Work on GSI-001, courtesy of A. Ghosh

Cable coupling vs. cable size Cable width Loss per unit length and kA Qc’ (Rc)/Icable Icable4 Qc’(Ra)/Icable Icable2 Size does matter, and small is better Dstrand = 1 mm Jop = 400 A/mm2

Re-cap on cable coupling Cables have an inherent uncertainty related to local geometry and electric contact properties Small cables have a clear loss advantage Interstrand resistance targets for pulsed cables are in the range of Ra ≥ 100  (adjacent) to Rc ≥ 10 m (transverse) Cable coupling loss (per unit cable volume) can be estimated as (unipolar cycle 0-Bmax-0): At Bmax = 4.5 T and dB/dt = 1.5 T/s (for a cable with 40 strands of 0.45 mm diameter, Rc = 10 , Ra = 100 m, w = 9.3 mm, t = 0.8 mm, Lp = 70 mm) we have E = 33 mJ/cm3 of cable This leads in our prototypical magnet to an AC loss of 0.8 W/m (average over a 12 s cycle), i.e. 20 % of the allocated thermal load budget

Dynamic resistance At high pulse rate, coupling currents may reach the critical current, and the shielding filaments/strands saturate. With no transport current (ideal case), saturation occurs at: The AC loss tops-off to a maximum value (penetration loss) In the presence of a transport current (practical case), saturation occurs at: The coupling currents wiggle around the filaments/strands, interfering with the flow of transport current A dynamic resistance appears against the transport current flow, causing a large increase in AC loss ≈ [1 + (Iop/Ic)2]

Hysteresis loss in the ferromagnetic yoke Magnetic steel for DC accelerator magnets have coercivity of ≈ 100 A/m and hysteresis loss (bipolar 1.5 T cycle) in the range of ≈ 900 J/m3 (≈ 115 mJ/kg) The coercivity (and hysteresis loss) is reduced by Si addition and by orientation of the grains to ≈ 30 A/m and ≈ 250 J/m3 (≈ 30 mJ/kg) Losses for uni-polar cycles are typically 30…50 % of the loss for bi-polar cycles and same amplitude Magnetic steel Electrical steel Measurement on steel candidates for SIS-300

Losses in electrical iron Hysteresis loss as a function of maximum field for a cycle 0-Bmax-0 Q = A/ Bmaxn (J/kg) A = 30…60 n = 2…3 Loss estimate for magnet Best electrical steel, up to saturation: ≈ 30 (mJ/kg) mass: My = (Rout2 - Rin2) (kg/m) Loss/cycle: 0.03 * My (J/m) Assume a yoke mass of 1.25 ton/m (our prototypical magnet with a ≈ 4.5 T bore field), the iron loss is 37.5 J/m, corresponding to 2.5 W/m over a 12 s cycle, i.e. 50 % of the allocated thermal budget Measurement on Si-steel used in GSI-001

A digression on critical current index heating V = V0 (Iop/Ic)n Usually small, but cannot be neglected if the cable has a low n, or working at a high fraction of Ic (Nb3Sn) A Nb3Sn cable operating at 70 % of Ic with an n of 10 will generate 1 W/m of magnet Iop = 20 kA V0 = 0.1 V/cm 40 turns/pole

A digression on critical temperature index heating Voltage characteristic of an ITER Nb3Sn CICC operating at variable temperature V = V0 (Top/Tc)m Moderate temperature changes can affect the index heating to a large amount A Nb3Sn cable operating at 4.5 K and with an m of 7 doubles the resistive voltage every 0.5 K Temperature (K) 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Electric field (V/cm) 0.05 0.1 0.15 0.2 0.25 Tc = 7 K V0 = 0.1 V/cm m = 7 Courtesy of D. Cyazinski

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

AC loss values for the baseline SPS+ dipole design AC loss is strongly dependent on magnet bore field and aperture as well as the details of the cross section: REPETITA (conductor/magnet optimizations) IUVANT

AC loss scaling with magnet design parameters Loss in the superconducting coil Hysteresis in the superconducting filaments: PM ≈ dfil Jc Vcoil log (Bmax) 1/tramp Coupling (eddy) currents in superconducting strands and cables: PC ≈ f[(,D)  (w2,Lp,N,Ra,Rc)] Vcoil Bmax2 1/tramp Loss in (optimised) iron Pyoke ≈ Vyoke 1/tramp Vcoil and Vyoke depend on Bmax and Dcoil operation magnet design strand strand and cable

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

A (f)lower-power option for PS2 Normal-conducting PS2 dipole Super-conducting iron-dominated PS2 dipole Iron weight [tons] 10 Peak voltage [V] 34 Average AC loss power [W] 1.3 Iron weight [tons] 15 Peak voltage [V] 41 Resistive power [W] 27000 L. Bottura, R. Maccaferri, C. Maglioni, V. Parma, L. Rossi, G. de Rijk, W. Scandale, Conceptual Design of Superferric Magnets for PS2, EDMS 871183 Potential for saving 7 MW of the 15 MW estimated total power consumption of PS2 complex

Outline Hand-waving arguments to set ball-park figures and target heat loads (in fast ramped NbTi magnets) Internal loss mechanisms in SC magnets Scaling of losses with magnet design (based on a study for an SPS upgrade) How to breach scaling relations (based on a study for a PS upgrade) More hand-waving arguments in an attempt to draw a summary

Protection and Stability Summary - 1/2 It is always beneficial to minimize AC loss, compatibly with protection, stability (transient heat balance) and current distribution Current Distribution The tri-lemma of the optimum pulsed superconducting cable design (PERITUS DELINEANDI OPTIMORUM DUCTORUM) (courtesy of P. Bruzzone, ECOMAG-05) Pulsed Field Conductor … and cost ! Heat Balance Protection and Stability AC Loss

Summary - 2/2 The best compromise of AC loss, current distribution, heat transfer, and cost can only be found in conjunction with the specific needs of the accelerator system and magnet design A reasonable internal heat load target per unit length for future superconducting ring accelerator magnets is in the range of 5 W/m to 10 W/m Higher values are not economically interesting Lower values may bear too much complication in the cable design The above target may be largely exceeded, for specific applications and locations, and over short lengths