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

2.1 Eddy Current Method

Eddy Current Penetration Depth δstandard penetration depth aluminum (σ = 26.7  10 6 S/m or 46 %IACS) 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 Depth [mm] | F |

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

2.2 Impedance Measurements

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

Resonance VeVe R L VoVo C

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

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

2.3 Impedance Diagrams

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

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 I 1 I 2

Probe Coil Impedance V 2 V 1 I 1 I 2 L, L, L R e

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

Electric Noise versus Lift-off Variation “Horizontal” Impedance Component “Vertical” Impedance Component Normalized Resistance Normalized Reactance lift-off “physical” coordinatesrotated coordinates

Conductivity Sensitivity, Gauge Factor Frequency [MHz] Gauge Factor, F absolute normal

Conductivity and Lift-off Trajectories lift-off trajectories are not straightconductivity trajectories are not semi-circles Normalized Resistance Normalized Reactance κ lift-off conductivity finite probe size Normalized Resistance Normalized Reactance κ lift-off conductivity

2.4 Test Coil Impedance

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

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

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

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

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

Resistance and Reactance of an Infinite Solenoid with Conducting Core Normalized Radius, b/δ g-function real part imaginary part

Effect of Changing Coil Radius a (changes) b (constant) lift-off Normalized Resistance Normalized Reactance b/δ = κ = a lift-off 

Effect of Changing Core Radius b (changing) a (constant) lift-off Normalized Resistance Normalized Reactance   n = 4 κ = b lift-off

Permeability

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

Solid Rod versus Tube b a c Normalized Resistance very thin solid rod tube Normalized Reactance thick tube σ1σ1 σ2σ2 σ1σ1 σ2σ2

Wall Thickness b a c 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

Wall Thickness versus Fill Factor b a c 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

Clad Rod b a c 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

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)

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

2.5 Field Distributions

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)

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] Frequency [MHz] standard actual aiai

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] analytical finite element

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] Frequency [MHz] standard actual

Lateral Resolution ferrite-core pancake coil (a i = mm, a o = 1.25 mm, h = 3 mm) in Ti-6Al-4V experimental FE prediction Radial Spread, a s [mm] Frequency [MHz]