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 H j H ds j dA A B Faraday‘s Law E E ds B dA t t A E 0 E dA 0 A B 0 B dA 0 A Material propertiesB 0 r H j E G.Moritz, 'Eddy Currents', CAS es,Brug 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 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 j E H j H j () 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<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 Hz(x,t)=H0*S(x,t) Diffusion time constant With response function S(x,t) 1S x ( ,t ) dt and S(x,0)=0 and S(x,t→∞)=1 d 0 1.0 9 = 110 , 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 Hz(,) xH t (0 1 erf ) 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 H, x t 1t 2 erfc e 0,003 z (A/m) z 0,002 t H 0 0,001 s t 0,000 H(,)(,) xz t Hz x t 0,000 0,001 0,002 0,003 0,004 0,005 0,006 0,007 t (sec) H 0 H 0 t t d x H (,)z x t t 0 t 0 0, if t >>τd Field lag Hed() x Analytical solutions: Half space conductor (1D-approach) (4) 3. Transient sinusoidal field Hz(t) = H0*sin (ωt) s t H(,)(,)(,) x tz Hz x t z H x t stationary transient x x 2 H x(,)s t H sin( e t ) z 0 Harmonic skin pthde G.Moritz, 'Eddy Currents', CAS es,Brug 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<x<+d Step‐function field (1D) n x cos t 2 4 d 2d H(z , x ) t0 H 1 4 n n n1 e n2 2 n1 n( 12 ) n odd LC iron, 1mm , µr=1000 Step‐function field (2D) 1m sec 1 n x m y cos cos t 2 2 2a 2b 2 n m H( , x , y ) t H 1 4 n, m 4 / z 0 2 e n, m 2 2 n,m odd n1 m1 f (.) n m a b G.Moritz, 'Eddy Currents', CAS es,Brug 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 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 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 - he mostt common way of numerical eddy current calculation BA B A E A t t t A JE t 2 Field calculation BAJ 2 A A Find vector potential t Diffusion equation for Vector ialenttpo A Sometimes the current vector potential T is d:eus j T Refer: MULTIMAG ‐ program for calculating and optimizing magnetic 2D and 3D fields in roaccelerat magnets (Alexander Kalimov [email protected]])[k 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 ay of eddy current calculationw 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 l •d,h<<penetration depth s y x (magnetically thin!) d •Steady state: t>>τd h ‐d/2 0 +d/2 neglecting the resistance From Faraday´s law (integral form) B z jy() x x contribution of the ends, since 2d<<l 2 2 From Ampere‘s law: B z d Heddy() x x2 z 2 4 Top/bottom of a G.Moritz, 'Eddy Currents', CAS es,Brug rectangular beam pipe! June 16 ‐ 25 2009 And then for the loss dP l 2 2 dPj xy() A l jy () x h dx in an area A=hdx: A d 2 3 2 2 lh d 2 1 d After integration P2 B dP orz P/ volume 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 B 2 dl orP 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 Copper 17.4 10‐9 0.156 10‐9 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.