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

2. 2.1 Eddy Current Method

3. Eddy Current Penetration Depth δstandard penetration depth aluminum (σ = 26.7  10 6 S/m or 46 %IACS) -0.2 0 0.2 0.4 0.6 0.8 1 0123 Depth [mm] Re { F} f = 0.05 MHz f = 0.2 MHz f = 1 MHz f = 0.05 MHz f = 0.2 MHz f = 1 MHz -0.2 0 0.2 0.4 0.6 0.8 1 0123 Depth [mm] | F |

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

5. 2.2 Impedance Measurements

6. Impedance Measurements IeIe VpVp ZpZp VeVe ZeZe VpVp ZpZp Voltage divider: Current generator: IeIe

7. Resonance VeVe R L VoVo C

8. Wheatstone Bridge VeVe V2V2 Z1Z1 Z4Z4 Z2Z2 Z3Z3 + _ G R 0 reference resistance L c reference (dummy) coil inductance R c reference coil resistance L * complex probe coil inductance probe coilreference coil

9. Impedance Bandwidth R 0 = 100 Ω, R c = 10 Ω

10. 2.3 Impedance Diagrams

11. Examples of Impedance Diagrams Im(Z) Re(Z) L C Im(Z) Re(Z) 0 Ω-Ω- Ω+Ω+ ∞ L C R 0 Ω-Ω- Ω+Ω+ ∞ R Im(Z) Re(Z) R L C 0 Ω ∞ R Im(Z) Re(Z) R2R2 L C 0 Ω ∞ R1R1 R1+R2R1+R2 R1R1

12. Magnetic Coupling I 1 N 1 N 2 V 2  11 V 1 I 2  22  12  21, V 1 V 2 L, L, L 111222 I 1 I 2

13. Probe Coil Impedance V 2 V 1 I 1 I 2 L, L, L 111222 R e

14. Impedance Diagram lift-off trajectories are straight: conductivity trajectories are semi-circles

15. Electric Noise versus Lift-off Variation 0.32 0.34 0.36 0.38 0.40 0.42 0.280.30.320.340.360.38 “Horizontal” Impedance Component “Vertical” Impedance Component 0.32 0.34 0.36 0.38 0.40 0.42 0.280.30.320.340.360.38 Normalized Resistance Normalized Reactance lift-off “physical” coordinatesrotated coordinates

16. Conductivity Sensitivity, Gauge Factor 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 00.20.40.60.81 Frequency [MHz] Gauge Factor, F absolute normal

17. Conductivity and Lift-off Trajectories lift-off trajectories are not straightconductivity trajectories are not semi-circles 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.5 Normalized Resistance Normalized Reactance κ lift-off conductivity finite probe size 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.10.20.30.40.5 Normalized Resistance Normalized Reactance κ lift-off conductivity

18. 2.4 Test Coil Impedance

19. Air-core Probe Coils single turnL = aL = 3 a acoil radius Lcoil length

20. Infinitely Long Solenoid Coil for outside loops (r 1,2 > a) for inside loops (r 1,2 < a) for encircling loops (r 1 < a < r 2 ) inside loop outside loopencircling 2a2a L + J s _ Js_ Js z

21. Magnetic Field of an Infinite Solenoid with Conducting Core in the air gap (b < r < a)H z = J s in the core (0 < r < b)H z = H 1 J 0 (k r) J n nth-order Bessel function of the first kind + J s _ Js_ Js 2 a 2 b z

22. Magnetic Flux of an Infinite Solenoid with Conducting Core + J s _ Js_ Js 2 a 2 b z

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

24. Resistance and Reactance of an Infinite Solenoid with Conducting Core 0.010.11101001000 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Normalized Radius, b/δ g-function real part imaginary part

25. Effect of Changing Coil Radius a (changes) b (constant) lift-off Normalized Resistance Normalized Reactance 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.5 b/δ = 1 3 5 10 20 2 κ = 1 0.9 0.8 0.7 a lift-off 

26. Effect of Changing Core Radius b (changing) a (constant) lift-off Normalized Resistance Normalized Reactance 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.5 100 400 9 25   n = 4 κ = 1 0.9 0.8 0.7 b lift-off

27. Permeability

28. Solid Rod versus Tube b a solid rod BC1: continuity of H z at r = b tube BC1: continuity of H z at r = b BC2: continuity of H z at r = c BC3: continuity of E φ at r = c b a c

29. Solid Rod versus Tube b a c 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.50.6 Normalized Resistance very thin solid rod tube Normalized Reactance thick tube σ1σ1 σ2σ2 σ1σ1 σ2σ2

30. Wall Thickness b a c 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.50.6 η = 0 solid rod b/  = 3 b/  = 2 Normalized Resistance Normalized Reactance b/  = 5 b/  = 10 b/  = 20 η  1 thin tube η = 0.2 η = 0.4 η = 0.6 η = 0.8

31. Wall Thickness versus Fill Factor b a c 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.50.6 Normalized Resistance Normalized Reactance solid rod κ = 0.95, η = 0 solid rod κ = 1, η = 0 thin tube κ = 1, η = 0.99 thin tube κ = 0.95, η = 0.99

32. Clad Rod b a c 0 0.2 0.4 0.6 0.8 1 00.10.20.30.40.50.6 Normalized Resistance Normalized Reactance copper cladding on brass core solid copper rod solid brass rod brass cladding on copper core d master curve for solid rod d thin wall lower fill factor

33. 2D Axisymmetric Models b a c 2ao2ao 2ai2ai t h ℓ short solenoid (2D) ↓ long solenoid (1D) ↓ thin-wall long solenoid (≈0D) ↓ coupled coils (0D) pancake coil (2D) Dodd and Deeds. J. Appl. Phys. (1968)

34. Flat Pancake Coil (2D) a 0 = 1 mm, a i = 0.5 mm, h = 0.05 mm,  = 1.5 %IACS,  =  0

35. 2.5 Field Distributions

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

37. Axial Penetration Depth air-core pancake coil (a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V Axial Penetration Depth, δ a [mm] 10 -2 10 -1 10 0 10 1 Frequency [MHz] 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 standard actual aiai

38. Radial Spread air-core pancake coil (a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V Radial Spread, a s [mm] Frequency [MHz] 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 analytical finite element 0.8 1.2 1.6 2.0 1.0 1.4 1.8

39. Radial Penetration Depth air-core pancake coil (a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V Radial Penetration Depth, δ r [mm] 10 -2 10 -1 10 0 10 1 Frequency [MHz] 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 standard actual

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