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2 Eddy Current Theory

2 Eddy Current Theory. 2.1 Eddy Current Method 2.2 Impedance Measurements 2.3 Impedance Diagrams 2.4 Test Coil Impedance 2.5 Field Distributions. 2.1 Eddy Current Method. 1. 1. f = 0.05 MHz. f = 0.05 MHz. 0.8. 0.8. f = 0.2 MHz. f = 0.2 MHz. 0.6. 0.6. f = 1 MHz. f = 1 MHz.

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2 Eddy Current Theory

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  1. 2 Eddy Current Theory 2.1 Eddy Current Method 2.2 Impedance Measurements 2.3 Impedance Diagrams 2.4 Test Coil Impedance 2.5 Field Distributions

  2. 2.1 Eddy Current Method

  3. 1 1 f = 0.05 MHz f = 0.05 MHz 0.8 0.8 f = 0.2 MHz f = 0.2 MHz 0.6 0.6 f = 1 MHz f = 1 MHz 0.4 0.4 0.2 0.2 0 0 | F | Re { F} -0.2 -0.2 0 0 1 1 2 2 3 3 Depth [mm] Depth [mm] Eddy Current Penetration Depth δ standard penetration depth aluminum (σ = 26.7  106 S/m or 46 %IACS)

  4. primary magnetic flux primary (excitation) current secondary secondary (eddy) current magnetic flux Eddy Currents, Lenz’s Law

  5. 2.2 Impedance Measurements

  6. Ze Zp Vp Ve Impedance Measurements Current generator: Zp Vp Ie Voltage divider: Ie

  7. R C L Vo Ve 1 Q = 2 0.8 Q = 5 Q = 10 0.6 Transfer Function, |K| 0.4 0.2 0 0 1 2 3 Normalized Frequency, w/W Resonance

  8. Z1 Z4 + G Ve _ V2 Z2 Z3 Wheatstone Bridge probe coil reference coil R0 reference resistance Lc reference (dummy) coil inductance Rc reference coil resistance L* complex probe coil inductance

  9. 0.5 0.4 0.3 Transfer Function, | K0 | 0.2 Lc = 100 µH Lc = 20 µH 0.1 Lc = 10 µH 0 0 1 2 3 Frequency [MHz] Impedance Bandwidth R0 = 100 Ω, Rc = 10 Ω

  10. 2.3 Impedance Diagrams

  11. Im(Z) Im(Z) Im(Z) Im(Z) Re(Z) Re(Z) Re(Z) Re(Z) Ω- 0 ∞ Ω+ Examples of Impedance Diagrams Ω- L L R R 0 ∞ C C Ω+ R2 R R1 0 0 Ω Ω L L R1 R ∞ ∞ R1+R2 C C

  12. I I I I 2 2 1 1 V N N V 1 2 1 2 F F F F , 11 12 21 22 V V L , L , L 1 2 11 12 22 Magnetic Coupling

  13. I I 2 1 V V L , L , L R 1 2 11 12 22 e Probe Coil Impedance

  14. 1 0.9 0.8 Re=30 W 0.7 0.6 0.5 Normalized Reactance 0.4 Re=10 W 0.3 0.2 κ = 0.6 Re=5 W κ = 0.8 0.1 κ = 0.9 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance Impedance Diagram lift-off trajectories are straight: conductivitytrajectories are semi-circles

  15. 0.42 0.42 0.40 0.40 0.38 0.38 “Vertical” Impedance Component 0.36 0.36 0.34 0.34 0.32 0.32 0.28 0.28 0.3 0.3 0.32 0.32 0.34 0.34 0.36 0.36 0.38 0.38 “Horizontal” Impedance Component Electric Noise versus Lift-off Variation “physical” coordinates rotated coordinates lift-off lift-off Normalized Reactance Normalized Resistance

  16. 0.14 0.42 0.12 0.40 0.10 lift-off 0.38 0.08 Gauge Factor, F 0.06 0.36 Normalized Reactance 0.04 absolute 0.34 0.02 normal 0 0.32 0.28 0.3 0.32 0.34 0.36 0.38 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] Normalized Resistance Conductivity Sensitivity, Gauge Factor

  17. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 lift-off 0.5 0.5 κ 0.4 0.4 κ lift-off Normalized Reactance Normalized Reactance 0.3 0.3 0.2 0.2 0.1 0.1 conductivity conductivity 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 Normalized Resistance Normalized Resistance Conductivity and Lift-off Trajectories finite probe size conductivity trajectories are not semi-circles lift-off trajectories are not straight

  18. 2.4 Test Coil Impedance

  19. a coil radius L coil length Air-core Probe Coils single turn L = a L = 3 a

  20. z + Js _Js for inside loops (r1,2 < a) inside loop encircling outside loop L for outside loops (r1,2 > a) 2a for encircling loops (r1 < a < r2) Infinitely Long Solenoid Coil

  21. + Js _Js z 2 b 2 a Magnetic Field of an Infinite Solenoid with Conducting Core in the air gap (b < r < a) Hz = Js in the core (0 < r < b) Hz = H1J0(kr) Jnnth-order Bessel function of the first kind

  22. z 2 b 2 a Magnetic Flux of an Infinite Solenoid with Conducting Core + Js _Js

  23. Impedance of an Infinite Solenoid with Conducting Core For an empty solenoid (b = 0): Normalized impedance:

  24. 1.2 real part 1.0 imaginary part 0.8 0.6 g-function 0.4 0.2 0.0 -0.2 -0.4 0.01 0.1 1 10 100 1000 Normalized Radius, b/δ Resistance and Reactance of an Infinite Solenoid with Conducting Core

  25. 1 b/δ = 1 0.8 2 0.7 0.6 lift-off 0.8 Normalized Reactance 3 a 0.4 0.9 w κ = 1 5 0.2 10 20 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance Effect of Changing Coil Radius lift-off a (changes) b (constant)

  26. 1 0.8 lift-off a (constant) 0.6 0.4 b (changing) 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Effect of Changing Core Radius wn = 4 0.7 lift-off 0.8 Normalized Reactance 9 b 0.9 w κ = 1 25 100 400 Normalized Resistance

  27. 4 µr = 4 ωn = 0.6 3 1 ω µ 3 1.5 Normalized Reactance 2 2 1 1 0 0 0.2 0.4 0.6 0.8 1 1.2 Normalized Resistance Permeability

  28. a c b a b Solid Rod versus Tube solid rod BC1: continuity of Hz at r = b tube BC1: continuity of Hz at r = b BC2: continuity of Hz at r = c BC3: continuity of Eφ at r = c

  29. 1 a thick tube 0.8 σ1 c b 0.6 Normalized Reactance solid rod σ2 σ1 0.4 very thin tube σ2 0.2 0 Normalized Resistance 0 0.1 0.2 0.3 0.4 0.5 0.6 Solid Rod versus Tube

  30. 1 0.8 b/ = 2 η = 0 solid rod 0.6 b/ = 3 a Normalized Reactance η = 0.2 η = 0.4 η = 0.6 0.4 c η = 0.8 b/ = 5 b 0.2 b/ = 10 b/ = 20 η 1 thin tube 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance Wall Thickness

  31. thin tube κ = 0.95, η = 0.99 solid rod κ = 1, η = 0 1 solid rod κ = 0.95, η = 0 0.8 Normalized Reactance a 0.6 c b 0.4 thin tube κ = 1, η = 0.99 0.2 Normalized Resistance 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Wall Thickness versus Fill Factor

  32. 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Clad Rod master curve for solid rod lower fill factor solid brass rod Normalized Reactance a d brass cladding on copper core copper cladding on brass core solid copper rod d b c thin wall Normalized Resistance

  33. 2ao 2ai h ℓ t Dodd and Deeds. J. Appl. Phys. (1968) 2D Axisymmetric Models a pancake coil (2D) c b short solenoid (2D) ↓ long solenoid (1D) ↓ thin-wall long solenoid (≈0D) ↓ coupled coils (0D)

  34. 0.2 1 coil diameter 4 mm 0.8 0.15 2 mm 1 mm 0.6 lift-off 0.1 fM 0.1 mm Normalized Reactance (Normal) Gauge Factor 0.4 0.05 mm 0.05 0.2 frequency 0 mm 0 0 0.1 1 10 100 0 0.05 0.1 0.15 0.2 0.25 0.3 Normalized Resistance Frequency [MHz] a0 = 1 mm, ai = 0.5 mm, h = 0.05 mm,  = 1.5 %IACS,  = 0 Flat Pancake Coil (2D)

  35. 2.5 Field Distributions

  36. electric field Eθ magnetic field (eddy current density) 10 Hz 10 kHz 1 MHz 10 MHz 1 mm air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm), in Ti-6Al-4V (σ = 1 %IACS) Field Distributions

  37. 101 standard actual 100 ai Axial Penetration Depth, δa [mm] 10-1 10-2 10-5 10-4 10-3 10-2 10-1 100 101 102 Frequency [MHz] air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V Axial Penetration Depth

  38. 2.0 analytical finite element 1.8 1.6 1.4 1.2 1.0 0.8 10-5 10-4 10-3 10-2 10-1 100 101 102 Frequency [MHz] air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V Radial Spread Radial Spread, as [mm]

  39. 101 standard actual 100 10-1 10-2 10-5 10-4 10-3 10-2 10-1 100 101 102 Frequency [MHz] air-core pancake coil (ai = 0.5 mm, ao = 0.75 mm, h = 2 mm) in Ti-6Al-4V Radial Penetration Depth Radial Penetration Depth, δr [mm]

  40. 1.8 FE prediction 1.6 experimental 1.4 1.2 1.0 Radial Spread, as [mm] 0.8 0.6 0.4 0.2 0 10-2 10-1 100 101 Frequency [MHz] ferrite-core pancake coil (ai = 0.625 mm, ao = 1.25 mm, h = 3 mm) in Ti-6Al-4V Lateral Resolution

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