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: to avoid them / to minimize the unwanted effects. G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 appropriate design 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 Ampere‘s Law H j Faraday‘s Law B E t •Quasistationary approach •No excess charge E 0 B 0 Material properties B 0 r H G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 H ds j dA A E ds t A B dA E dA 0 A B dA 0 A j E 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 magnetic flux." 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 H j j E H j ( ) H 2 H t 1 B E t 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‘) H z 2H z 2 t x Application of external field: Hz(t) Solution Hz(x,t) = ? Boundary conditions: Hz(0,t)=0 t<0 Hz(0,t)= Hz(t) t≥0 Hz(x,0)= 0 0<x<∞ 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 Diffusion time constant Hz(x,t)=H0*S(x,t) With response function S(x,t) and S(x,0)=0 and S(x,t→∞)=1 d 1 S ( x, t ) dt 0 1.0 9 0.8 = 110 , 0 H0 (A/m) 0.6 0.4 0.5 cm 1 cm 2 cm 0.2 0.0 x 2 t Similarity variable -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t (sec) Hz ( x , t ) H 0 (1 erf ) Special response function S(x,t) 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 0,006 Hz (A/m) 0,005 excitation 0.1 cm 0.2 cm 0.3 cm H0 2 2 2 H z x, t t 1 2 erfc e t0 0,004 0,003 0,002 H ( x, t ) H ( x, t ) 0,001 0,000 0,000 0,001 0,002 0,003 0,004 0,005 0,006 0,007 t (sec) s z t z H0 H0 t t x d H z ( x, t ) t0 t0 0, if t >>τd Field lag H ed (x) Analytical solutions: Half space conductor (1D-approach) (4) 3. Transient sinusoidal field Hz(t) = H0*sin (ωt) H z ( x, t ) H ( x, t ) H ( x, t ) s z t z stationary H ( x, t ) H 0 e s z transient x x sin(t ) 2 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 Hz(±d,t)=0 t<0 Hz(±d,t)= H0 t≥0 Hz(x,0)= 0 ‐d<x<+d Refer to Knoepfel 4.2, Table 4.2‐I Step‐function field (1D) nx cos t 2 d n H z ( x, t ) H 0 1 4 n 1 e n 1 n (1) 2 4 n n 2 2 d2 n odd LC iron, 1mm , µr=1000 1 1m sec Step‐function field (2D) nx my cos cos 2a 2b H z ( x, y, t ) H 0 1 4 2 f (n.m) n 1 m 1 n ,m e t n,m G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 2 n 2 m 2 4 / 2 2 n,m odd b a Analytical solutions: Field in the gap of an iron-dominated Cdipole For this special case: z y H l 2H 2H 0 x 2 y 2 g l t r G. Brianti et al., CERN SI/Int. DL/71‐3 (1971) g b l a x Case1 : g=0 Standard diffusion equation l g Case 2: Special diffusion equation r li H 1 H 2H 2H 0 2 2 g t 1 t y x 1 g 1 0 l 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 Analytical Methods cons physical understanding •simple geometry (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 B A Field calculation A B E A t t t A J E t 2 B A J A 2 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 [firstname.lastname@example.org]) 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 • assumptions •Field Bz only, uniform •d,h<<l •d,h<<penetration depth s (magnetically thin!) •Steady state: t>>τd 1. Eddy currents in a rectangular thin plate z l y x d B=(0,0, Bz) h ‐d/2 0 +d/2 j y ( x) From Faraday´s law (integral form) Bz x neglecting the resistance contribution of the ends, since 2d<<l 2 From Ampere‘s law: H eddy z Bz 2 d 2 x ( x) 2 4 G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Top/bottom of a rectangular beam pipe! And then for the loss dP in an area A=hdx: dP l 2 2 j y ( x) A l j y ( x) h dx A d 2 lh d 3 2 P 2 dP Bz 12 0 After integration Same formula as for a thin slab! 8000 SST, = 5E-7 m 6 Joule heat (watt/m) Joule heat (watt/m) 8 or formula FEM 4 2 CU, = 5E-10 m 6000 formula FEM 4000 1 d2 2 P / volume Bz 12 For Copper: No small perturbation anymore! 2000 0 0 0 1 2 3 4 0 5 1 2 3 4 dB/dt (T/sec) dB/dt (T/sec) 2. Eddy loss in a long,thin cylinder (radius r, length l, thickness d (r>>d)) j 5 r cos B r3 2 P B dl or (round thin beam pipe) 2 r P B 2 V 2 3. Eddy loss of round plate/disk (radius r, thickness d , r>>d) G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 2 r P B 2 V 8 Resistivity ρ (Ohm*m) @300K/4K (typical) 300K 4K LC steel (3% Silicon) 590 10‐9 440 x 10‐9 Stainless steel 720 10‐9 490 10‐9 Copper 17.4 10‐9 0.156 10‐9 G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Avoid copper! Eddy currents in magnets Different Magnet types WF Eddy currents in all conductive elements, especially in: •iron yoke (low carbon iron) •mechanical structure (low carbon iron • or stainless steel) •Coil (Copper, superconductor) • beam pipe (stainless steel Iron‐dominated magnets Coil dominated cos nθ‐ magnet G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 Laminated yoke (no isotropy!!) Recap: thin slab P~d2*σ Def.: Laminated magnets with insulated laminations and low conductivity!! Packing factor fp = Wi / (Wi + Wa ) Wi ‐ thickness single lamination Wa ‐ thickness of insulation Conductivity: σz=0 σxy≠0 Rel. permeability of laminations μr Effective permeability: µz, µxy FLUX DIRECTION z LAMINATION For laminations tangential to the flux: μeff = fp · (μr - 1 ) + 1 FLUX DIRECTION z LAMINATION For laminations normal to the flux: μeff = μr / (μr ‐ fp ∙ (μr ‐ 1)) Laminated yoke : µxy, µz • z , xy different for laminated magnets (highly anisotropic!!) μr ~ 4080 xy – xy = r(H) – z • fp = 1 : z = r(H) μ ~ 670 • fp < 1: z = 1 / (1 - fp) z = 15 – 50 μr → 1 r 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) Eddy losses: PW 2 2 Bpd 3 m 6 2 2 ν‐frequency (Hz) d ‐lamination thickness (m) ρ‐resistivity (Ohm*m) Bp‐ Induction amplitude (T) Hysteresis losses: from measurements wit a permeameter Anomalous losses = rest G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 P. Fabbriccatore, et al. Technical design Report SIS 300 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 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 z 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 20 t = 0.3 s 2 Current density, A/cm A.Kalimov, et al., IEEE Transactions on Applied Superconductivity., vol.12, No.1 pp. 98‐101, 2002 (MT17) 25 t = 1.1 s 15 t = 1.5 s t = 2.0 s 10 5 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 z, m Eddy current lines on the yoke backside surface. Yoke end . Current density distribution on the pole surface along the central line of the magnet 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) G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 R. Kurnyshov et al., Report on FE‐R&D, Contract No. 5, GSI, October 2008 Vectors of eddy current density in yoke vs time (µz=25) G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 R. Kurnyshov et al., Report on FE‐R&D, Contract No. 5, GSI, Field relaxation at different longitudinal positions SIS 100 dipole SIS100 Z=0 (Center of the magnet) Yoke -4 1.2x10 0.689 m -4 1.0x10 -4 3.0x10 1.17 m -5 -4 2.0x10 -5 6.0x10 -4 -5 4.0x10 1.0x10 By (T) By (T) 8.0x10 >0, max. 1G -5 2.0x10 Around 0, 0.0 -4 -1.0x10 0.00 By (T) SIS100 dipole •Linear ramp, up with 4 T/s, •1.9 T •All curves start at t=0 at the end of the ramp 0.05 0.10 0.15 0.20 0.25 -4 -2.0x10 time (sec) 0.00 0.05 0.10 1.4x10 0.20 0.25 -3 -1.0x10 1.01 m -3 1.2x10 -3 -2.0x10 -3 -3 -3.0x10 1.0x10 -4 8.0x10 >0, max. 14 G -4 6.0x10 1.319 m -3 -4.0x10 -3 -5.0x10 -3 -6.0x10 -4 4.0x10 <0, max.60 G -3 -7.0x10 -4 2.0x10 0.00 0.0 S. Y. Shim, to be published 0.15 time (sec) -3 By (T) 0.0 0.05 0.10 0.15 time (sec) 0.00 0.05 0.10 0.15 time (sec) 0.20 0.25 0.20 0.25 Transient field behaviour after a linear ramp y x 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 Fitting function Bt B 1 e t / 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! Flux‐loop minimized!! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009 M. Sorbi et al. Technical Design Report SIS 300 4.5T model dipole Eddy currents in resistive coil SIS 18 dipole A. Asner et al., SI/Int.DL/69-2 9.6.1969 (Booster Bending Magnet) Application of Biot‐Savart gives the eddy current contribution ito the field n the 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 Thickness 0.3 mm Average loss: 4.9 W/m (0-2T, 4T/s) For the use as cryopump: need to be cooled!) R. Kurnyshov et al., Report on FE‐R&D, Contract No. 5, GSI, 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 loss (W/m): 4.9 pipe only 8.1 with tubes 8.7 with tubes and ribs Current density (A/m2), end part 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 (µr>>1) 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.