EDDY CURRENT EFFECTS IN A PULSED FIELD MAGNETOMETER

Transcription

1 EDDY CURRENT EFFECTS IN A PULSED FIELD MAGNETOMETER R.Grössinger, M.Küpferling Institut. f. Festkörperphysik; Techn. Univ. Vienna; Austria

2 1) Introduction Two transient effects in magnetic materials: a) eddy currents in a conductor b) magnetic viscosity dh/dt [(GA/m)/s)] 20 Cu annealed 1.5 SmCo Cu Cu annealed 5-x x Cu 2.5 annealed Cu 3.0 annealed Cu 2.0 as-cast Cu 2.5 as-cast H [MA/m]

3 Eddy currents: Positive applications: Flux concentration Cnare effect achieves fields up to MG.

4 Eddy currents: Positive applications: Production of special metallic shapes. Sensoric metal Eddy currents

5 Eddy currents: Origin of eddy currents: Maxwell equations: Induction-law Electromagnetic wave Eddy currents

6 Eddy currents: a) eddy currents in the conductor b) eddy currents in the sample Origin of eddy currents:

7 a) Eddy current effects in the conductor: B [T] Field versus current through the thin foil magnet I [A] 200 V 500 V 750 V 1000 V 1200 V 1500 V 1750 V 2150 V

8 Field versus current through the thin foil magnet Note: field and current are not in phase! This changes also homogeneity of magnet! Changes zero-signal

9 a) Eddy current effects (?) at the beginning of the field pulse 0,6 M 0,4 0,2 dm/dt 0,0-0,2-0,4-0,6 0, , , , ,00020 t(s) Signal (V) of a N/N pick-up at the beginning of the field pulse (U C = 600V, C = 8 mf)

10 Reduction is achieved using inside of the pulse magnet a thin copper cylinder reduces this peak with a factor of about 30. 0,025 M 0,020 0,015 dm/dt 0,010 0,005 0,000-0,005 0, , , , ,00020 t(s) Signal (V) of N/N pick-up at beginning of field pulse (U C = 600V, C = 8 mf) - 50 Ohm resistance at end of measuring cables +Cu cylinder inside of pulse magnet.

11 b) Eddy current effects in the sample: M (A/m) Ni, cylindre, d=4mm, h=8mm ρ=8908kg/m 3 m=0,89183g annealed, 4h, 500 C 8mF, 2000V B (T)

12 2) Eddy current studies. Aim: Understanding of eddy currents correction of these effects. Comparison between experiment and modelling. Samples: spheres + cylinders and cuboids of Cu and Al Dimensions: Spheres: 4-10 mm Cylinders: length: 8mm, diameter 4mm After heat treatment (T = 500 C, t = 4h) Material specific resistivity Cu annealed µωm Al annealed µωm

13 Resistivity measurements at room temperature on same sample with same heat treatment! Eddy current measurements performed with different time constants. T.U.: T = 9.1 ms (8mF); T = 15.7 ms (24 mf) PFM: 40 ms and 57 ms (H max = 5T) Different time constants. Effect of sample geometry. Comparison of different materials

14 Comparison Cu - Al; annealed T = 9.1ms; units: A/m Cu-cylinder; Al-cylinder l = 8mm, O = 4mm annealed (T = 400 C, t = 2h) t = 9.1ms (C = 8mF) Cu Al A/m µ H (TESLA)

15 Cu - annealed; two different time constants: T = 9.1ms, T = 15.7 ms Cu-cylinder; l = 8mm, O = 4mm annealed (T = 400 C, t = 2h) T = 9.1 ms and T = 15.7 ms ms 9.1 ms A/m µ H (TESLA)

16 Cu-cylinder; T = 9.1 ms Cu, cylindre l=8mm, d=4mm heattreated 4h, 500 C m=0,90926g C=8mF, U=2000V magnetization (A/m) field (T)

17 Eddy current magnetization of annealed Cu and Al versus dh/dt Cu cylinder annealed T = 9.1 ms M versus dh/dt M(A/m) Al-cylinder; l =8mm; O = 4mm annealed (T = 400 C, t = 2h) T = 9.1ms ,00E+009-2,50E+009 0,00E+000 2,50E+009 5,00E dh/dt (A/ms)

18 Why scales M(eddy current) with dh/dt? Consequence of Maxwell equations: v r j = σ E. r r B = curla 1 r curl( curla ) = µ r curle r E r A = curl t r A = t. r r A 1 r j = σ = curl B t µ v j σ B = µ t

19 Experimetal proof of scaling of eddy current with dh/dt: µ H j = 0 ρ t M cylinder = j max. S µ 0 4 r = H ρ t r S 4 M M cylinder cylinder ( Cu) ( Al) = ρ ρ AL Cu M M ( Cu) cylinder AL = 2. 2 cylinder ( Al) = 2.3 ρ ρ Cu Slope proportional to spec. resistivity!

20 8,00E+013 6,00E+013 Cu, 24mF Zylinder, geom. Faktor=r p 4h(π/4) Kugel, geom. Faktor=r p 5(4π/15) m/geom.faktor (A/m 3 ) 4,00E+013 2,00E+013 0,00E+000-2,00E+013-4,00E+013-6,00E+013-2,00E+009-1,00E+009 0,00E+000 1,00E+009 2,00E+009 dh/dt (A/ms) Geometry corrected magnetic moment of a Cu sphere and a cylinder; pulse duration 15.7 ms; as a function of dh/dt

21 3) FE - calculations Finite element package FE by David Meeker [ Complete set of tools for solving static and low frequency 2D or axisymmetric problems in electrodynamics. Aim: understanding of results.

22 Theoretical considerations: Induced magnetic moment Cylindrical samples (r,h) M(eddy currents): r s h 2 µ dr' dz j( r') r' π = 0 0 (1) j(eddy currents) directly proportional to the radial distance r: jr (') j max r s r' =. (2)

23 Theoretical considerations: Induced magnetic moment Cylindrical samples (r,h) M(eddy currents): r s h 2 µ dr' dz j( r') r' π = 0 0 (1) j(eddy currents) directly proportional to the radial distance r: jr (') j max r s r' =. (2)

24 Magnetic moment Magnetization s j µ = h drr π = j hπ r max 3 '' max r s r 0 3 s 4. (4) M µ/v, (5) V...volume of cylinder V = r 2 s π h (6) Eddy current magnetization in a cylinder: M j r s = max 4 (A/m) (7)

25 After a similar calculation one finds: Eddy current magnetization of a sphere: M = jmaxrs 5 (A/m) Eddy current magnetization in a cylinder: M j r s = max 4 (A/m) How looks j as a function of r?

26 Cylinder: Eddy currents as a function of r: 3.000e e+002 J_eddy, MA/m^2 Re[J_eddy], MA/m^2 Im[J_eddy], MA/m^ e e e e e Length, mm

27 Sphere: Eddy currents as a function of r: 2.000e e+002 J_eddy, MA/m^2 Re[J_eddy], MA/m^2 Im[J_eddy], MA/m^ e e e Length, mm

28 Sphere: eddy currents for large r: 2.000e e+002 J_eddy, MA/m^2 Re[J_eddy], MA/m^2 Im[J_eddy], MA/m^ e e e Length, cm

29 eddy current density (MA/m^2) ,00 0,67 1,34 2,02 2,69 3,36 4,03 4,71 radius (cm) numerical solution J_eddy MA/m^2 numerical solution Re[J_eddy] MA/m^2 numerical solution Im[J_eddy] MA/m^2 analytical solution J_eddy MA/m^2 analytical solution Re[J_eddy] MA/m^2 analytical solution Im[J_eddy] MA/m^2

30 eddy current density (MA/m^2) 0,1 0,08 0,06 0,04 0,02 0-0,02-0,04-0,06 numerical solution J_eddy MA/m^2 numerical solution Re[J_eddy] MA/m^2 numerical solution Im[J_eddy] MA/m^2 analytical solution J_eddy MA/m^2 analytical solution Re[J_eddy] MA/m^2 analytical solution Im[J_eddy] MA/m^2-0,08 0,00 0,03 0,07 0,10 0,13 0,17 0,20 0,24 0,27 0,30 0,34 0,37 radius (cm)

31 4) Results Based on this assumption several cylindrical samples of Cu were studied. M (A/m) µ 0 H (T) Cu, cylinder, h=8mm d=2mm d=6mm d=8mm d=9,8mm d=4mm 8mF, T=9.1ms

32 Figure shows comparison of hysteresis loops of Cu-cylinders of different diameters; The magnetization increases with increasing sample diameter. Measured maximum magnetization fitted with a function f=cr 2, r is the radius of the sample as shown. The quadratic dependence results from the following formula: M = m V 1 max sample c 4 M scales therefore with r 2 = j r

33 If j max is a linear function of r sample, then M is a quadratic function of the sample radius. M scales therefore with r Cu, cylinder, h=8mm 8mF, 9.1ms M max (A/m) d (mm)

34 Comparison of hysteresis loops of Cu-cylinders of different heights; The magnetization is nearly constant Cu, cylinder, d=4mm h=2mm h=4mm h=6mm h=10mm h=8mm 8mF, T=9.1ms M (A/m) µ 0 H (T)

35 The maximum magnetization as a function of the height of the cylinder; The independence from h can be explained with the same formula as before: M = m V = j r 1 max sample c 4 If j max is independent from h, then M is constant for every cylinder with the same diameter but different height.

36 Dependence of the maximum magnetization as a function of the cylinder height M max (A/m) Cu, cylinder, d=4mm 8mF, 9.1ms h (mm)

37 5) Comparison of experimental (Mexp) and FE-results (Mnum): sample frequency Bmax (T) Mexp (ka/m) Mexp/f j_eddy_max Mnum (ka/m) Mnum/f Cu ,603 0,00 228,66 114,33 1,02 Cu1 109,89 5,603 0,00 222,93 111,47 1,01 Cu , ,84 213,59 106,80 0,95 Cu1 109,89 5, ,86 209,57 104,79 0,95 Cu2 109,89 5, ,82 202,94 101,47 0,92 Cu3 109,89 5,17 245,13 2,23 368,47 403,47 3,67 Cu2 63,69 5,17 48,8 0,77 116,90 58,45 0,92 Cu3 63,69 5,17 133,2 2,09 213,79 234,10 3,68 Al ,23 43,335 0,39 70,79 35,40 0,32 Al1 109,89 5,23 43,335 0,39 69,51 34,75 0,32 Al2 109,89 5, ,34 93,53 46,77 0,43 Al3 109,89 5, ,46 116,27 87,20 0,79 Al2 63,69 5,17 20,9 0,33 54,21 27,11 0,43 Al3 63,69 5,17 28,2 0,44 67,54 50,65 0,80 Cu1,Cu2,Al1,Al2: cylinders, h = 8mm, D = 4 mm Cu3 sphere, diameter 7.3mm, Al3 sphere, diameter 5mm Cu1 ρ = µωm, Al1 ρ = µωm Cu2,Cu3 ρ = µωm Al2,Al3 ρ = µωm

38 40 30 Cu, sphere, d=7.3mm, T=15.7ms calculated with FEMM measured 20 m (10-3 Am 2 ) µ 0 H (T) Comparison of the measured and calculated hysteresis loop of a Cu-sphere.

39 6) Eddy currents in magnetic, conducting materials M (A/m) Ni, cylindre, d=4mm, h=8mm ρ=8908kg/m 3 m=0,89183g annealed, 4h, 500 C 8mF, 2000V B (T)

40 Eddy currents in magnetic, conducting materials Ni cylinder 54 emu/g External field (koe)

41 J J error J error dymanic dh 1 dt dh 2 dt = J 7) A simple eddy current correction - the f-2f method static + 1,3 1 2 J J (Tesla) 1,35 1,25 1,2 1,15 1, Frequency (Hz) eddy _ error

42 M [T] NdFeB, cylinder h=6.9mm, d=20mm B r =1.25 T f-2f corrected, H c =1.603 T long pulse, H c =1.675 T short pulse, H c =1.696 T H [T] Hysteresis loop of a sintered Nd-Fe-B magnet (Vacodym 510) as measured with a f and a 2f pulse and applying the so-called f/2f correction.

43 8) Conclusion Good agreement between FE eddy current calculations and experimental results. Understanding of scaling j(eddy current) and M(eddy current) with dh/dt - for not too large r. Only harmonic 2D solution - general 3D solution much more complex. FE description of permanent magnet - nonlinear permeability - open!