Eddy currents (in accelerator magnets)
G. Moritz, GSI Darmstadt CAS Magnets, Bruges, June 16‐25 2009 Introduction
• Definition According to Faraday‘s law a voltage is induced in a conductor loop, if it is
subjected to a time-varying flux.As a result current flows in the conductor, if there exist a closed path. ‚Eddy currents‘ appear, if extended conducting media are subjected to time varying fields. They are now distributed in the conducting media.
• Effects • Field delay (Lenz ´s Law), field distortion
• Power loss
• Lorentz-forces
– Beneficial in some applications (brakes, dampers, shielding, induction heating, levitated train etc.)
– Mostly unwanted in accelerator magnets: appropriate design to avoid them / to minimize the unwanted effects.
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Outline
• Introduction – Definition, Effects (desired, undesired) • Basics – Maxwell-equations – Diffusion approach • Analytical solutions: Examples • Numerical solutions: Introduction of numerical codes – Direct application of Maxwell-equations (small perturbation) • Eddy currents in accelerator magnets – Yoke, mechanical structure, resistive coil, beam pipe • Design principles / Summary • Appendix (references)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Basics Maxwell-equations •Quasistationary approach •No excess charge
Ampere‘s Law jH dAjdsH A B Faraday‘s Law E dsE dAB t t A E 0 dAE 0 A B 0 dAB 0 A
Material properties 0 r EjHB
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Lenz‘s law "the emf induced in an electric circuit always acts in such a direction that the current it
drives around a closed circuit produces a
magnetic field which opposes the change in
Lenz‘s Law: Reason for field delay slow diffusion process
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Field diffusion equation one way of eddy current calculation vanish Ej
jH jH )(
B 2 H H E t t 1 Magnetic diffusivity •Assumption: σ uniform in space •Diffusion equation does also exist also for current density j, magnetic Induction B and magnet Vector Potential A ! •Having solved the differential equation for H, the eddy current density j can be calculated by Ampere‘s Law and consequently the power loss P. Analytical solutions: Half-space conductor (1D-approach)(1) (following closely H.E. Knoepfel ‚Magnetic Fields‘)
2 H H z z x2 t
Application of Boundary conditions:
external field: Hz(t) Hz(0,t)=0 t<0 Solution H (x,t) = ? z Hz(0,t)= Hz(t) t≥0
Hz(x,0)= 0 0 1. Step‐function field Hz(t) =H0=constant 2. Transient linear field Hz(t) = H0/t0*t 3. Transient sinusoidal field Hz(t) = H0*sin (ωt) G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Analytical solutions: Half-space conductor (1D-approach) (2) 1. Step‐function field Hz(t) =H0=constant Hz(x,t)=H0*S(x,t) Diffusion time constant With response function S(x,t) ),(1 dttxS and S(x,0)=0 and S(x,t→∞)=1 d 0 1.0 = 1109, 0 0.8 0.6 0.4 (A/m) 0 H 0.2 0.5 cm 1 cm 2 cm 0.0 x -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t (sec) 2 t H txz ),( 0 erfH )1( Special response function S(x,t) Similarity variable G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Analytical solutions: Half-space conductor (1D-approach) (3) 2. Transient linear field Hz(t) = H0/t0*t 0,007 excitation 0,006 0.1 cm 0.2 cm 0,005 0.3 cm 2 0,004 H 0 2 2 txH t erfc21, e 0,003 z (A/m) z 0,002 t H 0 0,001 s t 0,000 z z txHtxH ),(),( 0,000 0,001 0,002 0,003 0,004 0,005 0,006 0,007 t (sec) H 0 H 0 t d xt z txH ),( t 0 t 0 0, if t >>τd Field lag ed xH )( Analytical solutions: Half space conductor (1D-approach) (4) 3. Transient sinusoidal field Hz(t) = H0*sin (ωt) s t z z z txHtxHtxH ),(),(),( stationary transient x x 2 s ),( teHtxH )sin( z 0 Harmonic skin depth G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Skin depth as function of frequency and conductivity Refer: Knoepfel, fig. 4.2‐5 G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Analytical solutions: Slab conductor (lamination) Boundary conditions:for step function field H (±d,t)=0 t<0 z Refer to Knoepfel 4.2, Table 4.2‐I Hz(±d,t)= H0 t≥0 Hz(x,0)= 0 ‐d Step‐function field (1D) xn cos t 2 4 d 2d z HtxH 0 41),( n n n1 e n 22 n1 n )1( 2 n odd LC iron, 1mm , µr=1000 Step‐function field (2D) msec1 1 xn ym cos cos t 2 2 2a 2b 2 n m HtyxH 41),,( ,mn /4 z 0 2 e ,mn 2 2 n,m odd n1 m1 mnf ).( a b G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Analytical solutions: Field in the gap of an iron-dominated C- dipole 2 2 For this special H H l H z 2 2 0 case: x y l t g r y G. Brianti et al., CERN SI/Int. DL/71‐3 (1971) g Case1 : g=0 Standard diffusion equation x l g b Case 2: Special diffusion equation l r a 2 2 H H li H 1 H 2 2 0 x y g t 1 t 1 g 1 0l With this diffusivity κ1 we can use the slab solutions! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Analytical vs. numerical methods pros cons Analytical Methods physical •simple geometry understanding (mainly1D/ 2D) •Homogeneous, isotropic and linear materials •simple excitation Numerical Methods •complex geometry long computing (3D) times •inhomogeneous, anisotropic and nonlinear materials •complex excitation G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Vector Potential A - the most common way of numerical eddy current calculation AB B A E A tt t A EJ t 2 Field calculation JAB 2 A A Find vector potential t Diffusion equation for Vector potential A Sometimes the current vector potential T is used: j T Refer: MULTIMAG ‐ program for calculating and optimizing magnetic 2D and 3D fields in accelerator magnets (Alexander Kalimov [[email protected]]) Widely used numerical codes for the calculation of eddy current in magnets • Opera (Vector Fields Software, Cobham Techn. Services, Oxford) www.vectorfields.com – FEM – Opera 2d,AC and TR, Opera 3d, ELEKTRA , (TEMPO-thermal and stress-analysis) • ROXIE (Routine for the Optimization of Magnet X-Sections, Inverse Field Calculation and Coil End Design) (S. Russenschuck, CERN) https://espace.cern.ch/roxie/default.aspx – BEM/FEM – Optimization of cosnθ-magnets, coil coupling currents only • ANSYS (ANSYS Inc.) www.ansys.com/ – Finite Element Method – Direct and in-direct coupled analysis (Multiphysics) • eddy current heat rising temperature change resistivity change eddy current – “This feature is important especially in the region of cryogenic temperature. Because most of physical parameters depend highly on temperature in that region” . G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Direct application of maxwell equations -another way of eddy current calculation Eddy Current effect handled as small perturbation: •geometrical dimension << skin depth (high resistivity!) •low field ramp rate B=(0,0, Bz) 1. Eddy currents in a rectangular thin plate • assumptions •Field Bz only, uniform z •d,h< l •d,h< d •Steady state: t>>τd h ‐d/2 0 +d/2 neglecting the resistance From Faraday´s law (integral form) B z y xj )( x contribution of the ends, since 2d< 2 2 From Ampere‘s law: B z d eddy xH )( x2 z 42 Top/bottom of a G.Moritz, 'Eddy Currents', CAS Bruges, rectangular beam pipe! June 16 ‐ 25 2009 And then for the loss dP l 2 2 dP y )( y )( dxhxjlAxj in an area A=hdx: A d 2 3 2 2 dlh 2 1 d After integration 2 dPP z / volumePorB B z 0 12 12 8 8000 Same formula as for a thin SST, = 5E-7 m slab! CU, = 5E-10 m 6 6000 formula FEM formula FEM 4 (watt/m) 4000 For Copper: No small 2 2000 perturbation anymore! Joule heat (watt/m) Joule heat 0 0 012345 012345 dB/dt (T/sec) dB/dt (T/sec) 2. Eddy loss in a long,thin cylinder (radius r, length l, (round thin beam pipe) thickness d (r>>d)) r cos r3 r 2 j B P 2 ordlB P B 2 V 2 3. Eddy loss of round plate/disk (radius r, thickness d , r>>d) 2 r 2 G.Moritz, 'Eddy Currents', CAS Bruges, P B June 16 ‐ 25 2009 V 8 Resistivity ρ (Ohm*m) @300K/4K (typical) 300K 4K LC steel (3% Silicon) 590 10‐9 440 x 10‐9 ‐9 ‐9 Stainless steel 720 10 490 10 ‐9 ‐9 Copper 17.4 10 0.156 10 Avoid copper! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Eddy currents in magnets Different Magnet types WF Eddy currents in all conductive elements, especially in: •iron yoke (low carbon iron) Iron‐dominated •mechanical structure (low carbon iron magnets • or stainless steel) •Coil (Copper, superconductor) • beam pipe (stainless steel G.Moritz, 'Eddy Currents', CAS Bruges,Coil dominated cos nθ‐ magnet June 16 ‐ 25 2009 Laminated yoke (no isotropy!!) Laminated magnets with insulated Recap: thin slab 2 P~d *σ laminations and low conductivity!! Packing factor fp = Wi / (Wi + Wa ) W ‐ thickness single lamination Def.: i Wa ‐ thickness of insulation Conductivity: σz=0 σxy≠0 FLUX Rel. permeability of laminations μr DIRECTION Effective permeability: µz, µxy z FLUX DIRECTION LAMINATION LAMINATION z For laminations tangential to the flux: For laminations normal to the flux: μeff = fp ·(μr - 1 ) + 1 μ = μ / (μ ‐ f ∙ (μ ‐ 1)) eff r r p r Laminated yoke : µxy, µz • z , xy different for laminated μ ~ 4080 magnets (highly anisotropic!!) r xy – xy = r(H) – z μr → 1 • fp = 1 : z = r(H) μr ~ 670 • f < 1: = 1 / (1 - fp) p z z = 15 – 50 Electrical Yoke steel 3414 μr ~ 20 z for fp =0.95 μr ~ 15 μr ~ 10 μr → 1 (Courtesy of E. Fischer) Laminated yoke : choice of iron Low carbon silicon steel reduces Eddy current losses due to higher resistivity Hysteresis losses due to lower coercivity but P. Shcherbakov et al., Design Report SIS 300 6T dipole (2004) Practical limit G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Iron losses (steel supplier) Note: Steel suppliers give typically total losses at 50 Hz Total losses = eddy losses (~ν2) + hysteresis losses (~ν) + anomalous losses (~ν1.5) ν‐frequency (Hz) 22 d ‐lamination thickness (m) Eddy losses: P W dB 22 m3 6 p ρ‐resistivity (Ohm*m) Bp‐ Induction amplitude (T) Hysteresis losses: from measurements wit a permeameter Anomalous losses = rest P. Fabbriccatore, et al. Technical design Report SIS 300 G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 4.5T model dipole Yoke design of pulsed magnets • 2D‐design (ideal) No Bz, only Bx, By – Appropriate lamination thickness d (practical limit 0.3 mm) – Low steel conductivity – Low coercivity (to reduce the hysteresis z losses) • 3D‐design exist Bz eddy currents in the lamination sheet surface – Yoke end region – Areas with low packing factor z G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Yoke end region Large Bz components (up to 2.5T depending on the magnet) Source of eddy currents near the magnet end GSI: SIS18 dipole Courtesy of F. Klos Results 25 20 A.Kalimov, et al., IEEE Transactions on Applied t = 0.3 s 2 Superconductivity., vol.12, No.1 pp. 98‐101, t = 1.1 s 2002 (MT17) 15 t = 1.5 s t = 2.0 s 10 5 Current density, A/cm density, Current 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 z, m Eddy current lines on the yoke backside . Current density distribution on the pole surface. surface along the central line of the magnet Yoke end En d of ramp 0.05% Field integral: difference between static and Aperture field induced by the eddy currents dynamic value. field integral max: 48000 Gs.m SIS100 sc dipole model – laminated yoke µr(X,Y) ↔ original B(H) curve µr(Z) = 15 eddy current power in laminated yoke (S. Koch, H. de Gersem, T. Weiland ( TU Darmstadt)) Eddy current power in the yoke along the yoke Vectors of BZ in yoke vs time (µz=25) R. Kurnyshov et al., Report on FE‐R&D, Contract No. 5, G.Moritz, 'Eddy Currents', CAS Bruges, GSI, October 2008 June 16 ‐ 25 2009 Vectors of eddy current density in yoke vs time (µz=25) R. Kurnyshov et al., G.Moritz, 'Eddy Currents', CAS Bruges, Report on FE‐R&D, June 16 ‐ 25 2009 Contract No. 5, GSI, Field relaxation at different longitudinal positions SIS 100 dipoleSIS100 Yoke 1.2x10-4 Z=0 0.689 m -4 (Center of the magnet) 1.0x10 -4 3.0x10 1.17 m 8.0x10-5 2.0x10-4 -5 (T) 6.0x10 y 1.0x10-4 B -5 4.0x10 (T) Around 0, y 0.0 >0, max. 1G B -5 2.0x10 -1.0x10-4 0.0 0.00 0.05 0.10 0.15 0.20 0.25 -2.0x10-4 time (sec) SIS100 dipole 0.00 0.05 0.10 0.15 0.20 0.25 •Linear ramp, up with 4 T/s, time (sec) -3 •1.9 T 1.4x10 -1.0x10-3 -3 1.01 m -3 •All curves start at t=0 at the 1.2x10 -2.0x10 -3 -3 end of the ramp 1.0x10 -3.0x10 -4 -4.0x10-3 1.319 m 8.0x10 (T) y B -3 (T) -4 -5.0x10 y 6.0x10 >0, max. 14 G B -3 -4 -6.0x10 4.0x10 -7.0x10-3 <0, max.60 G 2.0x10-4 0.00 0.05 0.10 0.15 0.20 0.25 0.0 time (sec) S. Y. Shim, to be published 0.00 0.05 0.10 0.15 0.20 0.25 time (sec) Transient field behaviour after a linear ramp y Fitting function x 1 eBtB t / Z CNAO Scanning Dipole Magnet max. center field: 0.3 T 500 T/s Lamination thickness: 0.3 mm z= 0 magnet center, z=22.0 cm: end of yoke Field delay Diffusion time constant Magnet center By_max 0.7 % of maximum operation field Time constant some milliseconds Remark: In a medical accelerator like CNAO the beam energy is varied in small time steps less then the diffusion time constant! Courtesy of S. Y. Shim SIS 300 Dipole- eddy currents(direction and current density in the magnet ends 4.5 T, 1T/s 3.0 T, 1T/s M. Sorbi, et al., “Electromagnetic Design of the Coil‐Ends for the FAIR SIS300 Model Dipole,” in Proc. ASC’08, Chicago, 2008 G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Variation of the packing factor along the magnet (Synchrotron dipole of the HIT-facility Heidelberg) Six block structure leads to •Variation of the DC‐field •Field reduction by eddy currents (induced by local Bz components) in the AC‐case Courtesy of F. Klos Eddy currents in mechanical structure – Brackets – Endplates – Collar pins, Collar keys, Rods – Shield, shell, Try to avoid closed flux loops ! (for example by welding seams at the pole) LHC dipole cold mass Example: eddy curents in the copper shield of a sc dipole R&D magnet GSI001 Bmod at 4T nominal field Power density (W/m3) in the copper shield (resistivity: 2.5 nm) at 4T nominal field and a ramp rate of 4T/s. Maximum power density: 270 kW/m3 Courtesy of H. Leibrock G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Eddy currents in Pins, Rods and Keys (SIS 300 dipole) If possible: insulate them to avoid closed flux looops! M. Sorbi et al. Technical Design Report SIS 300 4.5T model dipole Flux‐loop minimized!! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Eddy currents in resistive coil SIS 18 dipole Application of Biot‐Savart gives the eddy current contribution ito the field n the A. Asner et al., SI/Int.DL/69-2 9.6.1969 (Booster Bending Magnet) magnet gap This case: Eddy currents improve field quality!! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Elliptical beam pipe (SIS 100 dipole) simulation of the eddy current loss density in the beam pipe R. Kurnyshov et al., Thickness 0.3 mm Report on FE‐R&D, Average loss: 4.9 W/m (0-2T, 4T/s) Contract No. 5, GSI, For the use as cryopump: need to be cooled!) October 2008 Elliptical beam pipe (SIS 100 dipole) - complete 3D model with ribs and cooling pipe Designed to avoid closed flux loops Current density (A/m2), central part Cycle:0 ‐2T, 4 T/s Average 4.9 pipe only Current density (A/m2), end part loss (W/m): 8.1 with tubes 8.7 with tubes and ribs Courtesy of S. Y. Shim Summary: Pulsed magnet design principles (to minimize eddy current effects) • Insulated laminations • Choice of iron • Appropriate magnet design – Avoid saturation (µ >>1) r – Rogowski‐profile of the magnet pole ends – Slits in the end laminations – Non‐conductive material at the magnet ends – ‚long‘ magnets (also from the eddy current aspect!) • Appropriate design of the mechanical structure – Choice of materials (non‐conductive wherever possible) – Avoid ‚bulky‘ components – Avoid magnetic ‚flux loops‘ • Field Control (‚B‐Train‘) G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 References •Books •Heinz E. Knoepfel, ‘Magnetic Fields’, John Wiley and Sons, INC. , New York…., 2000 •Jack T. Tanabe, Iron dominated electromagnets: design, fabrication, assembly and measurements, WORLD Scientific 2005 •Y. Iwasa, ‘Case studies in superconducting magnets’, Plenum Press, New York and London, 1994 •Reviews •K. Halbach, ‘Some eddy current effects in solid core magnets’, Nuclear Instruments and Methods, 107 (1973), 529-540 •E.E. Kriezis et al., 'Eddy Currents: Theory and Applications', Proceedings of the IEEE, Vol. 80, NO. 10. October 1992, p.1599-1589 Acknowledgement: I am greatly indebted to S.Y. Shim for his help during the preparation of this talk.