Electric Circuits, Kirchhoff’s Laws Kirchhoff’s loop rule (voltage law): R R ∫ E i d ℓ = 0 � I 2 1 V V + 2 ∑ V = 1 0 i Є V V R _ 0 3 3 V 4 Є electromotive force R 4 V i potential drop on i th element Kirchhoff’s junction rule (current law): = ∫∫ D Q i d S R R I I enc 2 1 1 2 S I 4 + I = ∑ 0 Є R i _ 3 I i current flowing into a R junction from the i th branch 4 Circuit Analysis Kirchhoff’s Laws: + − = V V V 0 1 4 0 R R I I + − = 1 2 V V V 0 1 2 2 3 4 I 4 V V + V V V 1 2 1 2 4 − − = 0 Є V V V R _ 0 4 3 3 R R R 1 2 4 V V 3 2 − = 0 R 4 R R 2 3 Loop Currents: R R I I 1 2 1 2 I + − − = i R ( i i ) R V 0 4 1 1 1 2 4 0 + Є i i R _ 1 2 3 + − − = i R i R ( i i ) R 0 2 2 2 3 1 2 4 R 4 13
DC Impedance Matching R g + V R � V � g _ = = W QV P IV 2 V = = 2 = � P I V I R � � � � � R � V V R g g � I = and V = � � + + R R R R � � g g 2 V ξ R g � P = , where ξ = � + ξ 2 R (1 ) R g g 2 V − ξ dP 1 g � = d ξ R + ξ 3 (1 ) g 2 V g = = P when R R max � g 4 R g AC Impedance I I I dI 1 = = = V V L dt V V R I V V ∫ I dt C � � � V 1 V V � � � = = = = ω = = Z Z i L Z R � � � ω I i C I I { } � � ω + ϕ � ω � V e ϕ = = i ( t ) = i t = i V Re V V t ( ) V e V e V V V 0 0 0 0 { } � ω + ϕ � ω � I e ϕ � i ( t ) i t i = I t ( ) = I e = I e I = I Re I I I 0 0 0 0 � V � 0 � ϕ = = + = i Z R i X Z e Z � I 0 V � 0 2 2 = = + Z R X I 0 X -1 arg( ) Z = ϕ = ϕ - ϕ = tan Z V I R 14
AC Power real notation complex notation correspondence { } � i ( ω + ϕ t ) � ω � = ω + ϕ = = i t = I t ( ) I cos( t ) I t ( ) I e I e I Re I I 0 0 0 I { } � ω + ϕ � ω � V t ( ) = V cos( ω + ϕ t ) = i ( t ) = i t V = Re V V t ( ) V e V e V 0 V 0 0 1 1 { } � � � � � � = * * = P I t V t ( ) ( ) P = I t V ( ) ( ) t = I V P Re P 0 0 2 2 1 1 � I V e ϕ − ϕ = ϕ − ϕ = i ( ) P I V cos( ) P I V 0 0 I V 0 0 2 2 reminder: α + β = α β − α β cos( ) cos cos sin sin cos( α − β = ) cos α cos β + sin α sin β 1 1 α + β + α − β = α β cos( ) cos( ) cos cos 2 2 α = i α + α e cos i sin AC Impedance Matching � Z g V ≈ � � V � Z � g { } � = P Re P 2 ⎧ ⎫ V * { } ⎪ ⎪ 1 Re Z g � � � = * = P I V Re ⎨ ⎬ � � � * 2 2 + + ( Z Z )( Z Z ) ⎪ ⎪ ⎩ g � g � ⎭ ( ) � � * = = = − Z Z R R , X X g � g � g � 2 ⎧ − ⎫ V R i X ⎪ ⎪ g g g = P Re ⎨ ⎬ � 2 2 4 R ⎪ ⎪ ⎩ g ⎭ 2 V g = P max 8 R g 15
1.3 Maxwell's Equations Vector Operations ∂ ∂ ∂ ∇ = + + e e e Nabla operator: x y z ∂ ∂ ∂ x y z 2 2 2 ∂ ∂ ∂ ∇ 2 = ∇ ∇ = + + i Laplacian operator: 2 2 2 ∂ ∂ ∂ x y z ∂φ ∂φ ∂φ ∇φ = + + e e e Gradient of a scalar: x y z ∂ x ∂ y ∂ z ⎧ ∫ A i d ℓ ⎫ � ( ) ∇× i = Curl of a vector: A e lim ⎨ ⎬ S ⎩ S ⎭ S → 0 ⎛ ∂ ⎞ ⎛ ∂ ⎞ ∂ A ∂ ∂ A ∂ A ⎛ A A ⎞ A y y z x z x ∇× A = e − + e − + e − a ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x y z ∂ ∂ ∂ ∂ ∂ ∂ y z ⎝ z x ⎠ x y ⎝ ⎠ ⎝ ⎠ ⎧ A i d S ⎫ ∫∫ ∂ A ⎪ ⎪ ∂ A ∂ A y S x z ∇ = = + + i A lim ⎨ ⎬ Divergence of a vector: ∂ ∂ ∂ V x y z → V 0 ⎪ ⎪ ⎩ ⎭ 2 2 2 ∂ φ ∂ φ ∂ φ ∇ φ = 2 + + Laplacian of a scalar: 2 2 2 ∂ x ∂ y ∂ z 2 2 2 2 ∇ = ∇ + ∇ + ∇ Laplacian of a vector: A A e A e A e x x y y z z ∇× ∇× = ∇ ∇⋅ − ∇ 2 Vector identity: ( A ) ( A ) A 16
Maxwell's Equations Field Equations: ∂ D ∇× = + ∂ Ampère's law: H J t ∂ B ∇× = − ∂ Faraday's law: E t ∇ = i D q Gauss' law: ∇ = i B 0 Gauss' law: Constitutive Equations: = σ J E conductivity = ε D E permittivity B = μ H permeability ( ε 0 ≈ 8.85 × 10 -12 As/Vm) ε = ε ε 0 r (µ 0 ≈ 4 π × 10 -7 Vs/Am) μ = μ μ 0 r 1.4 Electromagnetic Wave Propagation 17
Electromagnetic Wave Equation ω ω = i t = i t Harmonic time-dependence: E E e and H H e 0 0 Maxwell's equations: ∂ B ∇× = − = − ωμ ∇× ∇× = − ωμ σ + ωε E i H ( E ) i ( i ) E ∂ t ∇× ∇× = − ωμ σ + ωε ( H ) i ( i ) H ∂ D ∇× = + = σ+ ωε H J ( i ) E ∂ t ∇× ∇× = ∇ ∇⋅ − ∇ 2 ( A ) ( A ) A 2 ∇ E = i ωμ σ + ωε ( i ) E ∇⋅ E = 0 ∇⋅ = H 0 ∇ 2 = ωμ σ + ωε H i ( i ) H ∇ 2 + 2 = Wave equations: ( k ) E 0 ∇ 2 + 2 = ( k ) H 0 2 = − ωμ σ + ωε k i ( i ) E e ω − = = 0 i ( t k x ) Example plane wave solution: E E e e y y y H e ω − = = 0 i ( t k x ) H H e e z z z Wave Propagation versus Diffusion 2 k = − ωμ σ + ωε i ( i ) k wave number Propagating wave in free space: ω E e ω − = 0 i ( t x c / ) k = E e y c 1 c = H e ω − = 0 i ( t x c / ) μ ε H e z 0 0 c wave speed Propagating wave in dielectrics: 1 c = = = ε c n d r μ ε ε c 0 0 r d n refractive index Diffusive wave in conductors: 1 i − δ ω − δ = x / i ( t x / ) = − ωμσ = − E E e e e k i 0 y δ δ 1 δ = − x / δ − ω − i ( t x / ) δ = H H e e e 0 z π μσ f δ standard penetration depth 18
Intrinsic Wave Impedance E e ω − = = 0 i ( t k x ) H e ω − 0 i ( t k x ) E E e e = = H H e e y y y z z z = − ωμ σ + ωε k i ( i ) ∂ D ∇× = + = σ+ ωε H J ( i ) E ∂ t ∂ H z ω − ∇× = − = i ( t k x ) H e ik H e e y 0 y ∂ x ωμ E i 0 η = = σ+ ωε H i 0 Propagating wave in free space: μ 0 η = ≈ Ω 377 0 ε 0 Propagating wave in dielectrics: μ η 0 0 η = ≈ ε ε n 0 r Diffusive wave in conductors: ωμ + i 1 i η = = σ σδ Polarization Plane waves propagating in the x -direction: i ( ω − t k x ) i ( ω − t k x ) = + = + E E e E e E e e E e e y y z z y 0 y z 0 z ω − ω − = + = i ( t k x ) + i ( t k x ) H H e H e H e e H e e z z y y z 0 z y 0 y E E y 0 z 0 η = = − 0 H H z 0 y 0 i φ φ = = i E E e y E E e z y 0 y 0 z 0 z 0 z z z E E E E z y y y E y linear polarization elliptical polarization circular polarization φ − φ = φ − φ = 0º (or 180º) 90º (or 270º) y z y z 19
Reflection at Normal Incidence y I medium II medium incident x reflected transmitted e ω − e ω + e ω − i0 i ( t k x ) r0 i ( t k x ) t0 i ( t k x ) E = E e E = E e E = E e I I II i y r y t y E E E e ω − e ω + e ω − i0 i ( t k x ) r0 i ( t k x ) t0 i ( t k x ) H = η e H = − η e H = η e I I II i z r z t z I I II Boundary conditions: η − η − + E = = = + = E ( x 0 ) E ( x 0 ) E E E r0 II I = = η y y i0 r0 t0 R + η E i0 II I − + = = = + = H ( x 0 ) H ( x 0 ) H H H i0 r0 t0 z z η E E E E 2 t0 i0 r0 t0 = = η II − = T + η η η η E i0 II I I I II Reflection from Conductors y I dielectric II conductor incident x transmitted reflected “diffuse” wave 1 δ = ≈ 0 π μσ f ωμ η i 0 η = << η = II I σ n η − η = II I ≈ − R 1 η + η II I • negligible penetration • almost perfect reflection with phase reversal 20
Axial Skin Effect y E F x e ω i t E = ( ) e 0 y F x e ω = i t H H ( ) e 0 z propagating wave diffuse wave − δ − δ = x / i x / F x ( ) e e x δ standard penetration depth 1 δ = dielectric (air) conductor π μσ f 1 magnitude Normalized Depth Profile, F 0.8 real part 0.6 0.4 0.2 0 -0.2 0 1 2 3 Normalized Depth, x / δ Transverse Skin Effect r current density current, I E = E J 0 ( k r ) z 0 2 a z 1 i conductor rod 2 = − ωμσ k = − k i δ δ 1 δ = I π μσ f = magnitude, J DC 2 π a k I 8 = E Normalized Current Density, J / J DC 0 πσ 2 a J k a ( ) 7 a / δ = 1 1 a / δ = 3 6 J n n th-order Bessel function a / δ = 10 5 of the first kind 4 3 I k a J ( k r ) 0 = J ( ) r z 2 2 π a 2 J k a ( ) 1 1 0 0 0.2 0.4 0.6 0.8 1 Normalized Radius, r / a 21
Transverse Skin Effect r V current density Z = = R + i X I current, I 2 a z conductor rod = Z R G k a ( ) 0 � � � = ρ = R 0 2 A σπ a 100 ξ ξ J ( ) Normalized Resistance, R / R 0 0 ξ = G ( ) 2 J ( ) ξ 1 10 R ∝ ω a = + lim G (1 i ) ≈ R R δ δ→∞ 2 a / 0 1 � lim R = σ π δ 2 a a / δ→∞ 0.1 0.01 0.1 1 10 100 Normalized Radius, a / δ 22
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 23
Eddy Current Penetration Depth E F x e ω = i t E ( ) e 0 y F x e ω i t H = H ( ) e 0 z − x / δ − i x / δ = F x ( ) e e δ standard penetration depth 1 δ = π f μσ aluminum ( σ = 26.7 × 10 6 S/m or 46 %IACS) 1 1 f = 0.05 MHz f = 0.05 MHz 0.8 0.8 f = 0.2 MHz f = 0.2 MHz f = 1 MHz f = 1 MHz 0.6 0.6 Re { F } | F | 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0 1 2 3 0 1 2 3 Depth [mm] Depth [mm] Eddy Currents, Lenz’s Law primary magnetic flux primary magnetic field (excitation) current probe coil conducting specimen secondary secondary (eddy) current eddy currents magnetic flux ∇× = Φ = μ Λ H J N I p p p p p ∂ d ∇× = −μ − E ( H H ) = − Φ − Φ V ( ) s p s s p s ∂ t dt = σ J E I ∝ σ V s s s s ∇× H = J Φ = μ I Λ s s s s s d V = − N dt ( Φ − Φ ) p p s V p = ω σ μ � Z ( , , , , ... ) probe I p 24
2.2 Impedance Measurements Impedance Measurements Current generator: ω V ( ) p ω = = K ( ) Z I p I e Z p V p I ( ) ω e Voltage divider: Z e V ( ) ω Z p p I e ω = = K ( ) V ω + V ( ) Z Z e e p V e Z p V p ω K ( ) V Z = Z e p − ω 1 K ( ) V 25
Resonance Z ( ) ω ω V ( ) p R o ω = = K ( ) ω + ω V ( ) R Z ( ) e p ω i L ω = Z ( ) V e L C V o p − ω 2 1 LC ω i L R / ω = K ( ) + ω − ω 2 1 i L R / LC ω i 1 Ω Q Q = 2 K ( ) ω = ω i 0.8 Q = 5 + − ω 2 Ω 2 1 / Transfer Function, | K | Ω Q Q = 10 0.6 1 Ω = LC 0.4 C R = = = Ω 0.2 Q R R C Ω L L 0 1 0 1 2 3 ω = Ω − 1 o 2 4 Q Normalized Frequency, ω/Ω Wheatstone Bridge Z 1 Z 4 ⎛ ⎞ ω V ( ) Z Z 2 2 3 ω = = − K ( ) G ⎜ ⎟ V ( ) ω Z + Z Z + Z + ⎝ ⎠ e 1 2 4 3 V e _ G Z Z V 2 2 3 = = V 0 if 2 Z Z 1 4 Z 2 Z 3 probe coil reference coil = + ξ Z Z 3 (1 ) = = Z Z R 2 1 4 0 = ω * + Z i L R ⎛ ⎞ 2 c Z (1 + ξ ) Z 3 3 ω = − K ( ) G ⎜ ⎟ + + ξ + R Z (1 ) R Z ⎝ ⎠ = ω + Z i L R 0 3 0 3 3 c c ω ≈ ω ξ K ( ) G K ( ) R 0 reference resistance 0 L c reference (dummy) coil inductance Z R 3 0 R c reference coil resistance K ( ) ω = 0 + 2 ( R Z ) 0 3 L * complex probe coil inductance 26
Impedance Bandwidth ≈ ω Z i L ω ≈ ω ξ K ( ) G K ( ) 3 c 0 ω L / R c 0 ξ = ξ ω σ μ � ( , , , ,...) K ( ) ω ≈ 0 + ω 2 1 ( L / R ) c 0 Z R 3 0 ω = R K ( ) 0 0 ω = 2 + ( R Z ) p 0 3 L c = ω + Z i L R 1 3 c c ω ≈ K ( ) 0 p 2 0.5 R ω = 0 Transfer Function, | K 0 | 0.4 1 2 L c 0.3 2 R 0 ω = 2 L L c = 100 µH c 0.2 L c = 20 µH 2 ω ≈ L c = 10 µH K ( ) 0.1 0 1,2 5 ω 0 2 = 4 0 1 2 3 ω 1 Frequency [MHz] B ω − ω 6 2 1 = = = B 2 or 120% r R 0 = 100 Ω , R c = 10 Ω ω ω + ω 5 c 2 1 2.3 Impedance Diagrams 27
Examples of Impedance Diagrams Im( Z ) Im( Z ) Ω - Ω - L L R R 0 0 Re( Z ) Re( Z ) ∞ ∞ C C Ω + Ω + Im( Z ) Im( Z ) R R 2 R 1 L L 0 Ω 0 Ω Re( Z ) Re( Z ) R ∞ R 1 ∞ R 1 + R 2 C C Magnetic Coupling I 2 I 1 Φ Φ 12 21 = = κ Φ Φ 22 11 V d 1 N 1 N 2 V 2 V = N dt ( Φ + Φ ) 1 1 11 12 d V = N dt ( Φ + Φ ) 2 2 21 22 Φ Φ Φ Φ , 11 12 21 22 ⎡ V ⎤ ⎡ L L ⎤ ⎡ I ⎤ 1 11 12 1 = ω i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V L L I ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 2 21 22 2 I 2 I 1 I L I L 1 11 2 22 Φ = Φ = 11 22 N N 1 2 I L I L V 1 V 2 L , L , L Φ = κΦ = κ 1 11 Φ = κΦ = κ 2 22 11 12 22 21 11 12 22 N N 1 2 N N 2 1 = κ = κ L L L L 21 11 12 22 N N 1 2 = = κ L L L L 12 21 11 22 28
Probe Coil Impedance I 1 I 2 V = − I R = i ω L I + ω i L I 2 2 e 12 1 22 2 − ω i L 12 = I I V 1 2 1 V 2 L , L , L R e + ω R i L 11 12 22 e 22 = ω + ω V i L I i L I 1 11 1 12 2 ω 2 2 L = ω + 12 V ( i L ) I ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ V L L I 1 11 1 1 11 12 1 + ω = ω R i L i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ e 22 V L L I ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 2 12 22 2 ω 2 2 L 2 = κ 2 11 22 L L L � 12 = ω + Z i L 12 coil 11 R + ω i L e 22 κ = κ � ( ) ω L � � � 22 = + ξ ω σ � = + κ 2 Z Z ref [1 ( , , )] Z i coil n + ω R i L e 22 V � 1 = Z coil ω − ω I L R i L 1 � 2 22 e 22 = + κ Z i n + ω − ω R i L R i L e 22 e 22 � ≈ ω Z i L ref 11 � 2 2 ω ω L Z L R � coil � 2 22 e 2 22 = = + ξ Z = κ + i (1 − κ ) Z i (1 ) n n ω 2 + ω 2 2 2 + ω 2 2 L R L R L 11 e e 22 22 Impedance Diagram ζ = ω L / R 22 e ζ ζ 2 � = = κ 2 � 2 R Re{ Z } = = − κ X Im{ Z } 1 n n n n 2 + ζ + ζ 2 1 1 = μ Δ = L 3 H, f = 1 MHz, R / R 10% lift-off trajectories are straight: 22 e e 1 = − ζ X 1 R n n 0.9 0.8 R e =30 Ω conductivity trajectories are semi-circles Normalized Reactance 0.7 2 2 0.6 ⎛ ⎞ ⎛ ⎞ κ 2 κ 2 2 + − + = R X 1 ⎜ ⎟ ⎜ ⎟ n n 0.5 2 2 ⎝ ⎠ ⎝ ⎠ 0.4 R e =10 Ω 0.3 = = lim R 0 and lim X 1 n n 0.2 κ = 0.6 ω→ ω→ R e =5 Ω 0 0 κ = 0.8 0.1 κ = 0.9 2 = = − κ lim R 0 and lim X 1 n n 0 ω→∞ ω→∞ 0 0.1 0.2 0.3 0.4 0.5 κ 2 κ 2 Normalized Resistance ζ= = ζ= = − R ( 1) and X ( 1) 1 n n 2 2 29
Electric Noise versus Lift-off Variation “physical” coordinates rotated coordinates 0.42 0.42 “Vertical” Impedance Component 0.40 0.40 Δ � Z ⊥ Normalized Reactance lift-off Z Δ � n lift-off n 0.38 0.38 0.36 0.36 0.34 0.34 0.32 0.32 0.28 0.3 0.32 0.34 0.36 0.38 0.28 0.3 0.32 0.34 0.36 0.38 Normalized Resistance “Horizontal” Impedance Component Conductivity Sensitivity, Gauge Factor Δ R R / R = R 0 (1 + F ε ) = Δ � � F / = μ = Ω Δ = ± Ω L 3 H, f = 1 MHz, R 10 , R 1 22 e e 0.42 0.14 0.12 0.40 lift-off Normalized Reactance 0.10 Gauge Factor, F 0.38 Δ � 0.08 Z n 0.06 0.36 Δ � Z ⊥ n 0.04 0.34 absolute 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 Normalized Resistance Frequency [MHz] � Δ � ⊥ Z Δ Z n = Δ n = Δ F F norm abs R / R R / R e e e e 30
Conductivity and Lift-off Trajectories L κ ≈ κ � ≈ σ ( ) R e A L κ ≈ κ σ ≈ σ ( , ) � R finite probe size e σ A ( ) conductivity trajectories are not semi-circles lift-off trajectories are not straight 1 1 0.9 0.9 0.8 0.8 Normalized Reactance Normalized Reactance 0.7 0.7 0.6 lift-off 0.6 κ 0.5 0.5 κ lift-off 0.4 0.4 0.3 0.3 0.2 0.2 conductivity conductivity 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance Normalized Resistance 2.4 Test Coil Impedance 31
Air-core Probe Coils a coil radius L coil length single turn L = a L = 3 a I d � d H = e × e = ∫ H i d s I � � r enc π 2 4 r I N I H = lim H = center center 2 a L L a / →∞ 2 I a = H axis 2 2 3/ 2 2( a + z ) Infinitely Long Solenoid Coil = J n I s z ∫ H i d s = I � enc _ J s + J s for inside loops ( r 1,2 < a ) − = LH ( ) r L H ( r ) 0 z 1 z 2 = H constant z encircling inside loop outside loop L for outside loops ( r 1,2 > a ) − = LH ( ) r L H ( r ) 0 z 1 z 2 H = 0 z 2 a for encircling loops ( r 1 < a < r 2 ) − = LH ( ) r LH ( r ) L J z 1 z 2 s I H ( ) r = J = n I = N L z 1 s 32
Magnetic Field of an Infinite Solenoid with Conducting Core z 1 i 2 ∇ 2 + 2 = = − ωμσ k = − _ J s ( k ) H 0 k i + J s δ δ ⎛ 2 ⎞ ∂ ∂ 1 + + 2 = k H 0 ⎜ ⎟ z 2 ∂ r r ∂ r ⎝ ⎠ in the air gap ( b < r < a ) H z = J s in the core (0 < r < b ) H z = H 1 J 0 ( kr ) J n n th-order Bessel function of the first kind J s H = 1 J 0 ( kb ) J ( k r ) 0 = 2 b H J J z s ( k b ) 0 2 a Magnetic Flux of an Infinite Solenoid with Conducting Core J ( k r ) 0 = H ( ) r J J z s ( k b ) z 0 _ J s + J s Φ = = μ ∫∫ B dA ∫∫ H dA z z b Φ = πμ + πμ 2 − 2 2 H ( ) r r dr ( a b ) J ∫ z s = H H ( ) r z z 0 = b H J 2 z s 2 2 Φ = πμ + − J [ ∫ J ( k r r dr ) a b ] s 0 J ( kb ) 0 0 H = 0 z ξ ξ ξ = ξ ξ ∫ J ( ) d J ( ) 0 1 2 b J k b ( ) 1 2 2 Φ = πμ + − J [ a b ] s k J ( k b ) 0 ξ 2 J ( ) 1 ξ = ξ g ( ) 2 b J ( ) ξ 0 2 a Φ = πμ 2 − 2 − J s { a b [1 g k b ( )]} 33
Impedance of an Infinite Solenoid with Conducting Core Φ = πμ 2 − 2 − J s { a b [1 g k b ( )]} = = ωΦ = = J nI , V i , V NV nLV s 1 L 1 1 V V L 1 = = 2 = ωπμ 2 − 2 − 2 Z n L i { a b [1 g kb ( )]} n L I J s For an empty solenoid ( b = 0): = ωπμ 2 2 = Z i a n L i X e e Normalized impedance: Z = = − κ 2 − Z i {1 [1 g kb ( )]} n X e 2 b κ 2 = ≈ is called fill-factor ( lift-off) 2 a Resistance and Reactance of an Infinite Solenoid with Conducting Core = − κ 2 − = + Z i {1 [1 g kb ( )]} R i X n n n 2 1 i ) b i ⎛ b ⎞ k = − ξ = = − k b (1 i 2 ξ = − 2 ⎜ ⎟ δ δ δ δ ⎝ ⎠ = −κ 2 = − κ 2 − R Im{ ( g k b )} X 1 [1 Re{ ( g kb )}] n n − Re{ ( g k b )} 1 = − = X 1 m R m n n Im{ ( g k b )} ≤ ≤ 0 Re{ ( g kb )} 1 − 0.4 ≤ Im{ ( g k b )} ≤ 0 1.2 1.0 real part 0.8 imaginary part g -function 0.6 0.4 0.2 0.0 -0.2 -0.4 0.01 0.1 1 10 100 1000 Normalized Radius, b / δ 34
Effect of Changing Coil Radius 2 = − κ − Z i {1 [1 g k b ( )]} n 1 b/ δ = 1 b κ = a 0.8 2 Normalized Reactance lift-off 0.7 0.6 a (changes) lift-off 0.8 3 a 0.4 0.9 ω b (constant) κ = 1 5 0.2 10 20 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance Effect of Changing Core Radius = − κ 2 − Z i {1 [1 g k b ( )]} n 1 b κ = a 0.8 ω n = 4 Normalized Reactance 0.7 lift-off a (constant) 0.6 lift-off 0.8 9 b 0.9 0.4 ω b (changing) κ = 1 25 0.2 100 400 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance ω δ 1 ω = , where ω = ( a = ) n 1 ω 2 σμ a 2 1 0 ≈ − − 2 X 1 m R m R n 1 n 2 n 35
Permeability 4 b κ = = 0.8 b a 2 2 Φ = πμ μ + πμ − 2 ∫ H ( ) r r dr ( a b ) J r 0 z 0 s µ r = 4 ω n = 0.6 3 0 Normalized Reactance 1 ω µ 3 = − κ 2 − μ Z i {1 [1 g bk ( )]} n r 1.5 2 2 ω ω = ω n 1 1 1 1 δ ω = = ( a ) 1 2 σμ μ a 2 0 r 0 0 0.2 0.4 0.6 0.8 1 1.2 Normalized Resistance Solid Rod versus Tube b solid rod Φ = πμ 2 + πμ μ + πμ 2 − 2 c H 2 ∫ H ( ) r r dr ( a b ) J 0 3 r 0 z 0 s c BC1: continuity of H z at r = b = + H H J ( k r ) H Y k r ( ) a z 1 0 2 0 + = BC1: H J ( kb ) H Y kb ( ) J 1 0 2 0 s + = BC2: H J ( k c ) H Y k c ( ) H b 1 0 2 0 3 ∇× = = σ H J E tube ∂ H z − = σ E BC1: continuity of H z at r = b ϕ ∂ r BC2: continuity of H z at r = c k H J k c BC3: continuity of E φ at r = c − + = [ ( ) H Y k c ( )] E ( ) c ϕ 1 1 2 1 σ = H H z 3 a ωμ 2 π = π i H c E ( )2 c c 0 3 ϕ = H H ( ) r z z k c c + = BC3: H J k c ( ) H Y k c ( ) H H = J 1 1 2 1 3 z s 2 b = H 0 z 36
Solid Rod versus Tube 1 thick tube 0.8 b c Normalized Reactance κ = = η = 1, σ 1 a b 0.6 a solid rod 0.4 σ 2 σ 1 c very thin b tube 0.2 σ 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance Wall Thickness 1 0.8 b / δ = 2 η = 0 b c Normalized Reactance κ = = η = 1, solid rod a b 0.6 a b / δ = 3 η = 0.2 η = 0.4 η = 0.6 0.4 c η = 0.8 b b / δ = 5 0.2 b / δ = 10 b / δ = 20 η ≈ 1 thin tube 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance 37
Wall Thickness versus Fill Factor 1 thin tube κ = 0.95, η = 0.99 solid rod b c κ = η = 0.8 , κ = 1, η = 0 a b Normalized Reactance a 0.6 solid rod κ = 0.95, η = 0 c 0.4 b 0.2 thin tube κ = 1, η = 0.99 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance Clad Rod c b 2 2 Φ = πμ + πμ + πμ − 2 ∫ H ( ) r r dr 2 ∫ H ( ) r r dr ( a b ) J core core clad clad 0 s 0 c = ≤ < H H J ( k r ) 0 r c core 3 0 core = + ≤ < H H J ( k r ) H Y k ( r ) c r b clad 1 0 clad 2 0 clad 1 d = b − c = b (1 − η ) master curve for solid rod 0.8 σ b c clad Normalized Reactance κ = , η = , Σ = σ a b core 0.6 lower fill factor solid d a brass rod brass cladding 0.4 on copper core copper cladding solid on brass core d copper rod 0.2 b c thin wall 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance 38
2D Axisymmetric Models Dodd and Deeds. J. Appl. Phys. (1968) a pancake coil (2D) 2 a o 2 a i c h b ℓ t short solenoid (2D) ↓ ∞ ωπμ 2 2 i N α I ( ) 0 = α α Z ∫ f ( ) d long solenoid (1D) 2 2 6 − α h ( a a ) o i 0 ↓ α a o thin-wall long solenoid ( ≈ 0D) α = I ( ) ∫ x J ( ) x dx 1 ↓ α a i coupled coils (0D) αμ −α −α −α + −α h ( h � ) � 2 r 1 f ( ) α = 2( α + h e − 1) + [ e − e ] αμ +α r 1 α 2 = α 2 − 2 = α 2 + ωμ μ σ k i 1 r 0 Flat Pancake Coil (2D) a 0 = 1 mm, a i = 0.5 mm, h = 0.05 mm, σ = 1.5 %IACS, μ = μ 0 1 0.2 coil diameter 4 mm 0.8 2 mm Normalized Reactance 0.15 (Normal) Gauge Factor 1 mm lift-off 0.6 f M 0.1 mm 0.1 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] + a a 1 o i = = δ ⇒ = a f M 2 2 π σμ a 39
2.5 Field Distributions Field Distributions electric field E θ magnetic field 2 2 = + H H H (eddy current density) r z 10 Hz 10 kHz 1 MHz 10 MHz 1 mm air-core pancake coil ( a i = 0.5 mm, a o = 0.75 mm, h = 2 mm), in Ti-6Al-4V ( σ = 1 %IACS) 40
Axial Penetration Depth air-core pancake coil ( a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V 10 1 standard Axial Penetration Depth, δ a [mm] 1 δ = actual π σμ f 10 0 a i 10 -1 10 -2 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 Frequency [MHz] 1 1/e point below the surface at r = a = ( a + a ) i o 2 Radial Spread air-core pancake coil ( a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V 2.0 analytical ≈ a 2 a 1 2 finite element 1.8 Radial Spread, a s [mm] 1.6 1.4 1.2 1.0 a ≈ 1.2 a 2 o 0.8 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 Frequency [MHz] z = 1/e point from the axis at the surface ( 0) 41
Radial Penetration Depth air-core pancake coil ( a i = 0.5 mm, a o = 0.75 mm, h = 2 mm) in Ti-6Al-4V 10 1 standard Radial Penetration Depth, δ r [mm] 1 δ = actual π σμ f ≈ a 1.2 a 2 o 10 0 10 -1 10 -2 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 Frequency [MHz] δ = − a a r s 2 Lateral Resolution ferrite-core pancake coil ( a i = 0.625 mm, a o = 1.25 mm, h = 3 mm) in Ti-6Al-4V 1.8 FE prediction 1.6 experimental 1.4 Radial Spread, a s [mm] 1.2 1.0 0.8 0.6 0.4 0.2 0 10 -2 10 -1 10 0 10 1 Frequency [MHz] 42
3 Eddy Current NDE 3.1 Inspection Techniques 3.2 Instrumentation 3.3 Typical Applications 3.4 Special Example 3.1 Inspection Techniques 43
Coil Configurations voltmeter voltmeter voltmeter oscillator oscillator ~ ~ ~ oscillator ~ Z o excitation excitation coil coil coil sensing coil Hall or GMR detector testpiece testpiece testpiece differential coils parallel coaxial rotated Remote-Field Eddy Current Inspection ferromagnetic pipe exciter coil sensing coil Remote Field Near Field Remote Field ln( H z ) low frequency operation (10-100 Hz) 1 δ = π f μ μ σ r 0 e − δ = z / Exponentially decaying eddy currents H H z z 0 propagating mainly on the outer surface cause a diffuse magnetic field that leaks both z on the outside and the inside of the pipe. 44
Main Modes of Operation single-frequency time-multiplexed multiple-frequency Signal Signal Time Time pulsed frequency-multiplexed multiple-frequency Signal Signal Time Time τ ≈ μσ � 2 D excited signal (current) detected signal (voltage) Nonlinear Harmonic Analysis single frequency, linear response ferromagnetic phase (ferrite, martensite, etc.) Signal B Time H nonlinear harmonic analysis Signal Time 45
3.2 Eddy Current Instrumentation Single-Frequency Operation V r low-pass filter A/D converter V q 90º phase driver low-pass oscillator shifter amplifier filter driver processor + V m impedances phase _ balance V-gain H-gain probe coil(s) = ω − ϕ = ω = ω V V cos( t ), V V cos( t ), V V sin( t ) m s s r o q o 1 [ ] = ω − ϕ ω = ϕ + ω − ϕ V V V cos( t ) V cos( t ) V V cos( ) cos(2 t ) m r s s o s o s s 2 display 1 [ ] V V = V cos( ω − ϕ t ) V sin( ω t ) = V V sin( ϕ ) + sin(2 ω − ϕ t ) m q s s o s o s s 2 V V o o = ϕ = ϕ V V V cos( ), V V V sin( ) m r s s m q s s 2 2 46
Nonlinear Harmonic Operation V r low-pass oscillator filter A/D converter V q 90º phase driver low-pass n divider amplifier shifter filter driver processor V m + impedances phase _ balance V-gain H-gain probe coil(s) = ω − ϕ + ω − ϕ + ω − ϕ + V V cos( t ) V cos(2 t ) V cos(3 t ) ... m s1 s1 s2 s2 s3 s3 V = ω o V V o cos( n t ) V V = V cos( ϕ ) display r m r s n s n 2 V = ω o V V o sin( n t ) = ϕ V V V sin( ) q m q s n s n 2 Specialized versus General Purpose Nortec 2000S system Agilent 4294A system* frequency range* 0.1 – 10 MHz 0.1-80 MHz probe coil three pencil probes single spiral coil relative accuracy ≈ 0.1-0.2% ≈ 0.05-0.1% frequency scanning manual electronic measurement time ≈ 50 minutes for 21 points ≈ 3 minutes for 81 points *high-frequency application 47
Probe Considerations sensitivity ferrite-core coil air-core coil high coupling low coupling flat air-core coil high coupling high coupling high coupling eddy current eddy current eddy current thermal stability I 2 I 1 I ⎡ V ⎤ ⎡ Z Z ⎤ ⎡ I ⎤ = 1 11 12 1 V Z I = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V V Z Z I V 2 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ V 1 2 12 22 2 = ω * + Z i L R wire = ω * Z i L 12 12 Φ 11 Φ 12 Φ 21 Φ 22 Φ , topology flexible, low self-capacitance, reproducible, interchangeable, economic, etc. 3.3 Eddy Current NDE Applications • conductivity measurement • permeability measurement • metal thickness measurement • coating thickness measurements • flaw detection 48
3.3.1 Conductivity Conductivity versus Probe Impedance constant frequency 1 Titanium, 6Al-4V 0.8 Inconel Normalized Reactance Stainless Steel, 304 0.6 Copper 70%, Nickel 30% 0.4 Lead Magnesium, A280 0.2 Nickel Aluminum, 7075-T6 Copper 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance 49
Conductivity versus Alloying and Temper IACS = International Annealed Copper Standard σ IACS = 5.8 × 10 7 Ω -1 m -1 at 20 °C ρ IACS = 1.7241 × 10 -8 Ω m 60 2014 2024 6061 7075 Conductivity [% IACS] 50 T0 T0 T0 T0 40 T6 T73 T72 T76 T4 T6 T8 T6 30 T3 T4 T6 T3 T4 20 Various Aluminum Alloys Apparent Eddy Current Conductivity 1.0 0.8 Normalized Reactance 0.6 lift-off curves magnetic field 0.4 conductivity probe coil (frequency) 0.2 curve specimen 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance eddy currents σ = σ 2 � = s Normalized Reactance • high accuracy ( ≤ 0.1 %) 4 σ = σ 1 • controlled penetration depth 3 σ, � 2 � = 0 1 Normalized Resistance 50
Lift-Off Curvature inductive capacitive (low frequency) (high frequency) lift-off lift-off ℓ = s ℓ = 0 ℓ = s ℓ = 0 σ 2 σ 2 “Vertical” Component “Vertical” Component . . conductivity conductivity σ σ σ 1 σ 1 “Horizontal” Component “Horizontal” Component Inductive Lift-Off Effect 4 mm diameter 8 mm diameter 2.0 2.0 1.5 %IACS 1.5 %IACS 1.5 1.5 63.5 μ m Relative Δ AECC [%] . Relative Δ AECC [%] . 1.0 1.0 50.8 μ m 38.1 μ m 0.5 0.5 25.4 μ m 0.0 0.0 19.1 μ m -0.5 -0.5 12.7 μ m -1.0 -1.0 6.4 μ m -1.5 -1.5 0.0 μ m -2.0 -2.0 0.1 1 10 100 0.1 1 10 100 Frequency [MHz] Frequency [MHz] 80 80 70 70 63.5 μ m 60 60 50.8 μ m AECL [ μ m] 50 . . 50 . AECL [ μ m] 38.1 μ m 40 40 25.4 μ m 30 30 19.1 μ m 20 12.7 μ m 20 6.4 μ m 10 10 0.0 μ m 0 0 -10 -10 0.1 1 10 100 0.1 1 10 100 Frequency [MHz] Frequency [MHz] 51
Instrument Calibration conductivity spectra comparison on IN718 specimens of different peening intensities 3.0 12A Nortec 8A Nortec 2.5 4A Nortec 12A Agilent 8A Agilent 2.0 AECC Change [%] 4A Agilent . 12A UniWest 1.5 8A UniWest 4A UniWest 1.0 12A Stanford 8A Stanford 4A Stanford 0.5 0.0 -0.5 0.1 1 10 100 Frequency [MHz] Nortec 2000S, Agilent 4294A, Stanford Research SR844, and UniWest US-450 3.3.2 Permeability 52
Magnetic Susceptibility paramagnetic materials with small ferromagnetic phase content moderately high susceptibility low susceptibility 4 1.0 permeability 0.8 Normalized Reactance µ r = 4 Normalized Reactance 3 permeability 3 lift-off 0.6 2 2 frequency 0.4 (conductivity) 1 frequency 1 (conductivity) 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance Normalized Resistance increasing magnetic susceptibility decreases the apparent eddy current conductivity (AECC) Magnetic Susceptibility versus Cold Work cold work (plastic deformation at room temperature) causes martensitic (ferromagnetic) phase transformation in austenitic stainless steels 10 1 SS304L SS302 10 0 SS304 Magnetic Susceptibility 10 -1 10 -2 SS305 IN718 10 -3 IN625 IN276 10 -4 0 10 20 30 40 50 60 Cold Work [%] 53
3.3.3 Metal Thickness Thickness versus Normalized Impedance scanning probe coil thickness loss due to corrosion, erosion, etc. 1 aluminum ( σ = 46 %IACS) 0.8 1 thinning Normalized Reactance f = 0.05 MHz 0.8 f = 0.2 MHz lift-off 0.6 f = 1 MHz 0.6 Re { F } 0.4 0.4 − δ − δ = x / i x / F x ( ) e e 0.2 thick thin plate plate 0 0.2 -0.2 0 1 2 3 0 Depth [mm] 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Resistance 54
Thickness Correction Vic-3D simulation, Inconel plates ( σ = 1.33 %IACS) a o = 4.5 mm, a i = 2.25 mm, h = 2.25 mm 1.4 Conductivity [%IACS] 1.3 thickness 1.0 mm 1.5 mm 1.2 2.0 mm 2.5 mm 3.0 mm 3.5 mm 1.1 4.0 mm 5.0 mm 6.0 mm 1.0 0.1 1 10 Frequency [MHz] 3.3.4 Coating Thickness 55
Non-conducting Coating probe coil, a o ℓ non-conducting t coating d conducting substrate a o > t, d > δ , AECL = ℓ + t a o = 4 mm, simulated a o = 4 mm, experimental lift-off: 80 80 70 63.5 μ m 70 60 60 50.8 μ m 50 50 AECL [ μ m] AECL [ μ m] 38.1 μ m 40 40 25.4 μ m 30 30 19.1 μ m 20 20 12.7 μ m 10 10 6.4 μ m 0 0 0 μ m -10 -10 0.1 1 10 100 0.1 1 10 100 Frequency [MHz] Frequency [MHz] Conducting Coating probe coil, a o ℓ conducting t coating z = δ e J e d z conducting substrate (µ s , σ s ) approximate: large transducer, weak perturbation equivalent depth: δ s δ = e 2 ⎛ ⎞ 1 ( ) ≈ σ δ = σ⎜ ⎟ AECC( ) f e ⎜ ⎟ 2 π f μ σ ⎝ ⎠ s s ⎛ ⎞ 1 σ ≈ ⎜ ⎟ ( ) z AECC ⎜ ⎟ 2 4 π z μ σ ⎝ ⎠ s s analytical: Fourier decomposition (Dodd and Deeds) numerical: finite element, finite difference, volume integral, etc. (Vic-3D, Opera 3D, etc.) 56
Simplistic Inversion of AECC Spectra 0.254-mm-thick surface layer of 1% excess conductivity 1.2 1.2 uniform Conductivity Change [%] 1 1 input profile AECC Change [%] 0.8 0.8 0.6 0.6 inverted from 0.4 0.4 AECC 0.2 0.2 0 0 -0.2 -0.2 0 0.2 0.4 0.6 0.8 1 0.001 0.1 10 1000 Depth [mm] Frequency [MHz] 1.2 1.2 Gaussian Conductivity Change [%] 1 1 input profile AECC Change [%] 0.8 0.8 0.6 0.6 inverted from 0.4 0.4 AECC 0.2 0.2 0 0 -0.2 -0.2 0.001 0.1 10 1000 0 0.2 0.4 0.6 0.8 1 Frequency [MHz] Depth [mm] 3.3.5 Flaw Detection 57
Impedance Diagram 1 0.8 conductivity (frequency) lift-off Normalized Reactance flawless 0.6 material ω 1 crack depth 0.4 ω 2 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Normalized Resistance apparent eddy current conductivity (AECC) decreases apparent eddy current lift-off (AECL) increases Crack Contrast and Resolution Vic-3D simulation a o = 1 mm, a i = 0.75 mm, h = 1.5 mm probe coil austenitic stainless steel, σ = 2.5 %IACS, μ r = 1 f = 5 MHz, δ ≈ 0.19 mm crack 1 -10% threshold 0.8 Normalized AECC 0.6 0.4 0.2 detection threshold 0 0 1 2 3 4 5 Flaw Length [mm] semi-circular crack 58
Eddy Current Images of Small Fatigue Cracks probe coil crack 0.5” × 0.5”, 2 MHz, 0.060”-diameter coil Al2024, 0.025-mil crack Ti-6Al-4V, 0.026-mil-crack Crystallographic Texture = σ J E generally anisotropic hexagonal (transversely isotropic) cubic (isotropic) σ σ σ ⎡ J ⎤ ⎡ 0 0 ⎤ ⎡ E ⎤ ⎡ J ⎤ ⎡ 0 0 ⎤ ⎡ E ⎤ ⎡ J ⎤ ⎡ 0 0 ⎤ ⎡ E ⎤ 1 1 1 1 1 1 1 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ J = 0 σ 0 E J = 0 σ 0 E J = 0 σ 0 E ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ 1 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ σ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ σ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ σ ⎥ ⎢ ⎥ J 0 0 E J 0 0 E J 0 0 E ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 3 3 3 3 2 3 3 1 3 x 1 θ x 3 σ M σ n σ m basal plane x 2 surface plane σ < σ σ 1 conductivity normal to the basal plane 1 2 σ 2 conductivity in the basal plane 2 2 σ θ = σ θ + σ θ ( ) cos sin n 1 2 θ polar angle from the normal of the basal plane 2 2 σ ( ) θ = σ sin θ + σ cos θ m 1 2 σ m minimum conductivity in the surface plane σ = σ M 2 σ M maximum conductivity in the surface plane σ θ = σ 2 θ + σ + 2 θ ( ) ½ [ sin (1 cos )] σ a average conductivity in the surface plane a 1 2 59
Electric “Birefringence” Due to Texture 500 kHz, racetrack coil highly textured Ti-6Al-4V plate equiaxed GTD-111 1.05 1.40 1.04 1.38 Conductivity [%IACS] Conductivity [%IACS] 1.03 1.36 1.02 1.34 1.01 1.32 1.00 1.30 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Azimuthal Angle [deg] Azimuthal Angle [deg] Grain Noise in Ti-6Al-4V 1” × 1”, 2 MHz, 0.060”-diameter coil as-received billet material solution treated and annealed heat-treated, coarse heat-treated, very coarse heat-treated, large colonies equiaxed beta annealed 60
Eddy Current versus Acoustic Microscopy 1” × 1”, coarse grained Ti-6Al-4V sample 5 MHz eddy current 40 MHz acoustic Inhomogeneity AECC Images of Waspaloy and IN100 Specimens inhomogeneous Waspaloy homogeneous IN100 4.2” × 2.1”, 6 MHz 2.2” × 1.1”, 6 MHz conductivity range ≈ 1.38-1.47 %IACS conductivity range ≈ 1.33-1.34 %IACS ±3 % relative variation ±0.4 % relative variation 61
Conductivity Material Noise as-forged Waspaloy 1.50 1.48 1.46 1.44 AECC [%IACS] 1.42 1.40 1.38 1.36 Spot 1 (1.441 %IACS) Spot 2 (1.428 %IACS) 1.34 Spot 3 (1.395 %IACS) 1.32 Spot 4 (1.382% IACS) 1.30 0.1 1 10 Frequency [MHz] no (average) frequency dependence Magnetic Susceptibility Material Noise 1” × 1”, stainless steel 304 0.51×0.26×0.03 mm 3 edm notch intact f = 0.1 MHz, Δ AECC ≈ 6.4 % f = 0.1 MHz, Δ AECC ≈ 8.6 % f = 5 MHz, Δ AECC ≈ 0.8 % f = 5 MHz, Δ AECC ≈ 1.2 % 62
3.4 Special Example Residual Stress Assessment 1500 Alternating Stress [MPa] 1000 with opposite residual stress service load 500 intact (no residual stress) endurance natural increased limit life time life time 0 10 2 10 4 10 6 10 8 Fatigue Life [cycles] Residual stresses have numerous origins that are highly variable. Residual stresses relax at service temperatures. 63
Surface-Enhancement Techniques Shot Peening (SP) Laser Shock Peening (LSP) Low-Plasticity Burnishing (LPB) 200 50 Ti-6Al-4V 0 Residual Stress [MPa] 40 SP Almen 4A Cold Work [%] SP Almen 12A -200 LSP 30 LPB -400 Ti-6Al-4V 20 -600 SP Almen 4A SP Almen 12A 10 LSP -800 LPB -1000 0 0 0.2 0.4 0.6 1.0 1.2 0.4 0 0.2 0.6 1.0 1.2 Depth [mm] Depth [mm] Piezoresistive Effect parallel, normal, circular F F Electroelastic Tensor: ⎡ Δσ σ ⎤ ⎡ κ κ κ ⎤ ⎡ τ ⎤ / / E δ 1 0 11 12 12 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Δσ σ = κ κ κ τ / / E ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 0 12 11 12 2 ⎢ Δσ σ ⎥ ⎢ κ κ κ ⎥ ⎢ τ ⎥ / / E ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 80 3 0 12 12 11 3 Axial Stress [ksi] 60 40 τ = τ = τ τ = 0 Isotropic Plane-Stress ( and ) : 20 1 2 ip 3 0 Δσ σ / -20 η = a 0 = κ + κ ip τ 11 12 -40 ip E / Time [1 s/div] Conductivity [%IACS] 1.403 IN 718, parallel 1.402 Adiabatic Electroelastic Coefficients: 1.401 * κ = κ + κ 1.4 11 11 th 1.399 * κ = κ + κ 1.398 12 12 th 1.397 Time [1 s/div] 64
Material Types Ti-6Al-4V Al 2024 Al 7075 0.004 0.004 0.004 parallel parallel parallel normal normal normal 0.002 0.002 0.002 Δσ / σ 0 Δσ / σ 0 Δσ / σ 0 0 0 0 -0.002 -0.002 -0.002 -0.004 -0.004 -0.004 -0.001 0 0.001 0.002 -0.002 0 0.002 0.004 -0.001 0 0.001 0.002 τ ua / E τ ua / E τ ua / E Waspaloy IN718 Copper 0.004 0.004 0.004 parallel parallel parallel normal normal normal 0.002 0.002 0.002 Δσ / σ 0 Δσ / σ 0 Δσ / σ 0 0 0 0 -0.002 -0.002 -0.002 -0.004 -0.004 -0.004 -0.002 0 0.002 0.004 -0.002 0 0.002 0.004 -0.001 0 0.001 0.002 τ ua / E τ ua / E τ ua / E XRD and AECC Measurements Waspaloy 500 50 3 Almen 4A Conductivity Change [%] Residual Stress [MPa] 0 40 Almen 8A 2 Cold Work [%] Almen 12A -500 30 Almen 16A 1 Almen 4A Almen 4A -1000 20 Almen 8A Almen 8A 0 Almen 12A Almen 12A -1500 10 Almen 16A Almen 16A -2000 0 -1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.1 1 10 Depth [mm] Depth [mm] Frequency [MHz] 50 500 3 Almen 4A Conductivity Change [%] Residual Stress [MPa] 40 0 Almen 8A 2 Cold Work [%] Almen 12A -500 30 Almen 16A 1 Almen 4A Almen 4A -1000 20 Almen 8A Almen 8A 0 Almen 12A Almen 12A 10 -1500 Almen 16A Almen 16A 0 -2000 -1 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.1 1 10 Depth [mm] Depth [mm] Frequency [MHz] before (solid circles) and after full relaxation for 24 hrs at 900 °C (empty circles) 65
Thermal Stress Relaxation in Waspaloy Waspaloy, Almen 8A, repeated 24-hour heat treatments at increasing temperatures 0.6 intact Apparent Conductivity Change [% ] 300 °C 0.5 350 °C 400 °C 0.4 450 °C 500 °C 550 °C 0.3 600 °C 650 °C 700 °C 0.2 750 °C 800 °C 850 °C 0.1 900 °C 0 0.1 0.16 0.25 0.4 0.63 1 1.6 2.5 4 6.3 10 Frequency [MHz] The excess apparent conductivity gradually vanishes during thermal relaxation! XRD versus Eddy Current inversion of measured AECC in low-plasticity burnished Waspaloy 1.2 20 200 eddy current XRD 0 1.0 15 Residual Stress [MPa] -200 0.8 AECC Change [%] Cold Work [%] . . -400 0.6 10 -600 0.4 -800 0.2 5 -1000 0.0 XRD -1200 eddy current -0.2 0 -1400 0.01 0.1 1 10 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 Frequency [MHz] Depth [mm] Depth [mm] 66
XRD versus High-Frequency Eddy Current shot peened IN100 specimens of Almen 4A, 8A and 12A peening intensity levels 40 200 Almen 4A (XRD) 0 Almen 8A (XRD) -200 Residual Stress [MPa] . 30 Almen 12A (XRD) -400 Cold Work [%] . -600 Almen 4A (AECC) 20 -800 Almen 8A (AECC) -1000 Almen 12A (AECC) -1200 Almen 4A (XRD) 10 -1400 Almen 8A (XRD) -1600 Almen 12A (XRD) -1800 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Depth [mm] Depth [mm] ≈ 50 MHz 67
4 Magnetic NDE 4.1 Magnetic Properties 4.2 Magnetic Measurements 4.3 Magnetic Materials Characterization 4.4 Magnetic Flaw Detection 4.1 Magnetic Properties 68
Magnetization p m magnetic dipole moment N number of turns = p N I A m I current A encircled vector area + I - I Q charge 1 = × p 2 Q R v v velocity m R radius vector M magnetization ∑ p m = M V volume V χ magnetic susceptibility = χ M H H magnetic field = μ + = μ μ B magnetic flux density B ( H M ) H 0 0 r μ 0 permeability of free space μ = + χ 1 r μ r relative permeability Classification of Magnetic Materials Diamagnetism: μ r < 1 no remanence orbit distortion e.g., copper, mercury, gold, zinc Paramagnetism: μ r > 1 no remanence orbit and spin alignment e.g., aluminum, titanium, platinum Ferromagnetism: μ r >> 1 remanence, coercivity, hysteresis self-amplifying paramagnetism Curie temperature e.g., iron, nickel, cobalt 69
Diamagnetism = + p p p m orb spin π 2 Q A e r v = = = − magnetic dipole moment p N I A p m orb τ π 2 r B p spin electron spin erv = − electron orbital motion p orb p orb 2 number of turns Q N v Φ d F e − = 2 π = − π 2 current r E r I i dt e F m encircled area A Φ = d m dv = 2 π F ev B r charge of proton m e dt e dt τ orbital period m 2 π = 2 π Δ B r r v orbital radius r e B orbital velocity v er Δ = v B induced electric field 2 E i m decelerating electric force F e 2 2 μ 2 2 v e r e r 0 Δ = − = − Q p B H orb mass of electron F e m 4 4 m m dipoles within unit volume = n F eE i μ 2 2 e e r 0 χ = − n χ magnetic susceptibility orb 4 m - χ ≈ 1-10 ppm Weak Paramagnetism, Curie Law = + p p p m orb spin = × B T p B m m = θ sin T p B m m F m θ θ p m = ( ) θ θ = sin θ θ ∫ ∫ U T d p B d m m - I θ � � 90 90 T m + I = − θ cos U p B m m F m = − p i U B m m magnetic dipole moment p m − U U − m m0 magnetic flux density B = ( ) k T p U e B m magnetic force F m twisting moment or torque T m potential energy of the dipole U m Curie Law: Boltzmann constant k B μ 2 M n m C 0 χ = = = absolute temperature T 3 H k T T B dipoles within unit volume n χ ≈ 5-50 ppm χ magnetic susceptibility 70
Strong Paramagnetism, Curie-Weiss Law: M χ = H C ≈ Curie law: M H T M magnetization C χ ≈ H exciting magnetic field T χ magnetic susceptibility = + = + α H H H H M C material constant t i T absolute temperature C = M H t H t total magnetic field T H i interaction field M M M χ = = = α material factor − M T H H H t i − α M C T C Curie temperature C χ = − α T C C χ = Curie-Weiss law: − T T C Ferromagnetism (i) magnetic polarization is produced by collective action of similarly oriented spins within magnetic domains (ii) very high permeability (iii) magnetic hysteresis (v) remnant magnetic polarization (remanence) (vi) coercive magnetic field (coercivity) (iv) depolarization above the (magnetic) Curie temperature B B r first magnetization H H c 71
Spontaneous Magnetization [001] [111] [010] “easy” magnetic axis [100] [110] N N N N N N S S N S N S S S S S S S N N S N S N = + + U U U U total internal wall external Magnetic Domains in Single Crystals easy magnetic axes 1 demagnetization H = 0 (spontaneous magnetization) domain wall movement B 4 5 2 partial magnetization 3 H 2 irreversible rotation H 1 3 “knee” of the H magnetization curve reversible rotation 4 technical saturation H thermal precession not shown 5 full saturation (no precession) 72
4.2 Magnetic Measurements Magnetic Sensors noise threshold 10 5 10 4 Hall Flux Density [pT/Hz 1/2 ] 10 3 GMR 10 2 SDP 10 1 10 0 fluxgate 10 -1 SQUID 10 -2 0 5 10 15 20 25 Frequency [Hz] Φ d V = − N = − ω i N AB coil: axial dt 73
Hall Detector z y x = + × F Q ( E v B ) B z = − + = F e E ( v B ) 0 y y x z V H = E y b a I I x x F e I = − enab v x x F m a I x V = a E = − av B = B H y x z z enb R I = H x V B H z V H b 1 = R H en Fluxgate B 1 hard magnetic cores low-frequency or dc high-frequency B B I exc external magnetic field excitation B 2 H sensing voltage V sens (to be low-pass filtered) B = 0 B ≠ 0 B 1 B 1 t t B 2 B 2 t t B 1 + B 2 B 1 + B 2 t t 74
Vibrating-Sample Magnetometer vibration ( ω ) V sens B 0 B 0 bias magnetic flux density M magnetization B 0 M = χ μ χ magnetic susceptibility 0 µ 0 permeability of free space d specimen displacement d = d 0 sin( ω t ) d 0 specimen amplitude Φ = + μ κ ω ( ) t A B [ M sin( t )] 1 0 0 ω angular frequency Φ t = A B − μ M κ ω t t time ( ) [ sin( )] 2 0 0 κ geometrical coupling factor ∂Φ ∂Φ 1 2 = − + V sens ( ) t N N A coil cross section ∂ ∂ t t Φ 1,2 flux in coil 1 and 2 N number of turns = − ωχ κ ω V ( ) t 2 N A B cos( t ) sens 0 V sens sensing voltage Faraday Balance electromagnet specimen W’ = W - F m spacer h U m magnetic potential energy precision scale p m magnetic dipole moment B magnetic flux density M magnetization = − p U i B for a single dipole: m m V volume U g gravitational potential energy = − for a given magnetized volume: U M V B m U total potential energy = + U U U g m h height = − U W h M V B W actual weight dU dB = = − W ' W M V W ’ apparent weight dh dh χ magnetic susceptibility = χ M H H magnetic field μ 2 dH V dH 0 W ' − W = − μ χ V H dh = − χ µ 0 permeability of free space 0 2 dh 75
4.3 Magnetic Materials Characterization Magnetic Properties = μ + B 0 ( H M ) para- and diamagnetic materials: = χ M H = μ μ B H 0 r μ = + χ 1 r = = μ + μ B B H M ( , ) H M H M ( , ) ferromagnetic materials: p 0 0 p 1.5 hardened steel 1 Flux Density [Tesla] 0.5 soft iron 0 -0.5 -1 -1.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Magnetic Field [kA/m] 76
Initial Magnetization anhysteretic initial magnetization curve Flux Density Flux Density Differential Permeability Magnetic Field B magnetic flux density dB H magnetic field μ = d dH M magnetization B = μ 0 ( H + M ) µ 0 permeability of free space µ d differential permeability lim M = M 0 H →∞ M 0 saturation magnetization M ≤ n p n dipoles per unit volume 0 m p m magnetic dipole moment Retentivity, Coercivity, Hysteresis = μ + B 0 ( H M ) = M M H M ( , ) p B technical saturation: B r H H H c remanence [Vs/m 2 ] = μ B r B M r 0 r M r remnant magnetization H + M H ( ) = 0 c c µ 0 permeability of free space = M H ( ) 0 ci H c coercive field [A/m] H ≤ H c ci H ci intrinsic coercivity = dU BdH 0 U 0 magnetic energy density Δ = U A hysteresis area [J/m 3 ] A 0 77
Texture, Residual Stress mild steel (Langman 1985) 2 2 B || B || σ = 0 MPa σ = 36 MPa 1 1 B ⊥ Flux Density [T] Flux Density [T] B ⊥ 0 0 -1 -1 -2 -2 -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 Magnetic Field [A/m] Magnetic Field [A/m] 2 2 σ = 110 MPa σ = 183 MPa B || B || 1 1 Flux Density [T] Flux Density [T] B ⊥ B ⊥ 0 0 -1 -1 -2 -2 -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 Magnetic Field [A/m] Magnetic Field [A/m] Magnetostriction Spontaneous magnetostriction: domain M = M ≤ M easy magnetic axes s 0 volume M ≈ 0 H = 0 ε domain = ε domain = e , 0 1 2,3 e volume ε = 1,2,3 3 Induced magnetostriction: 2 e ε = 1 3 ε e H 1 ε = − = − 2,3 2 3 ε − ε = e 1 2 M s spontaneous magnetization M 0 saturation magnetization e spontaneous strain within a single domain ε 1,2,3 principal strains 78
Barkhausen Noise B H = 0 domain wall H movement H magnetic field Barkhausen noise Amplitude • magnetic Barkhausen noise • acoustic Barkhausen noise Time Curie Temperature χ magnetic susceptibility C C material constant χ = Curie-Weiss law: − T T C T temperature T C Curie temperature ferromagnetic materials ( T < T C ): 1.2 1.0 typical alloy 0.8 M s / M 0 0.6 typical pure metal 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T / T C 79
4.4 Magnetic Flaw Detection Magnetic Flux Leakage exciter coil sensor (small coil, Hall cell, etc.) ferromagnetic test piece Advantages: Disadvantages: fast material sensitive inexpensive poor sensitivity large, awkward shaped specimens (particle) poor penetration depth 80
Magnetic Boundary Conditions Gauss' law: Ampère's law: ∇ = ∇× = i B 0 H J x n x n medium II medium II B II θ ΙΙ θ ΙΙ B II,n H II B I,t H I,t H II,n boundary x t x t B II,t H II,t B I,n B I θ Ι θ Ι H I,n H I medium I medium I = = B B H H I,n II,n I,t II,t μ = μ θ = θ H H tan H tan H I I,n II II,n I I,n II II,n θ θ tan tan I = II μ μ I II Magnetic Refraction B II θ θ tan tan I = II θ ΙΙ μ μ I II 90 Nonmagnetic Angle, θ II [deg] µ I /µ II = medium II 75 (air) 10 30 60 100 45 medium I θ Ι (ferromagnetic) 30 B I 15 0 B II θ ΙΙ 0 15 30 45 60 75 90 medium II Ferromagnetic Angle, θ I [deg] (air) B I medium I θ Ι (ferromagnetic) 81
Exciter Magnets air gap ferromagnetic core electromagnet H magnetic field H d � = N I = MMF ∫ � N number of turns Φ = μ μ r H A 0 I excitation current Φ = ∫ μ μ MMF magnetomotive force MMF d � � A 0 r Φ magnetic flux MMF ℓ length of flux line = R m Φ µ 0 µ r magnetic permeability 1 d � 1 � A cross section area i = ≈ ∑ R ∫ � m μ μ μ μ A A R m magnetic reluctance 0 r 0 i r i i Yoke Excitation N I Detection Methods: • magnetic particle electromagnet (gravitation, friction, adhesion, cohesion, magnetization) • magnetic particle with ultraviolet paint magnetometer • coil • Hall detector, GMR sensor crack • fluxgate, etc. Tangential Magnetic Field Normal Magnetic Field Lateral Position Lateral Position 82
Subsurface Flaw Detection B 2 1 H saturation greatly reduces the differential permeability low magnetic field high magnetic field crack crack 83
5 Current Field Measurement 5.1 Alternating Current Field Measurement 5.2 Direct Current Potential Drop 5.3 Alternating Current Potential Drop 5.1 Alternating Current Field Measurement 84
Principle of Operation magnetic injection: ~ ~ normal (z) primary ac flux transverse ( y ) axial ( x ) magnetometer electric field magnetic flux density galvanic current injection ≈ Field Perturbation normal (z) transverse ( y ) B z [a.u.] axial ( x ) axial scanning B z < 0 above flaw Axial Position B z [a.u.] magnetometer cw current magnetic flux electric density B x 0 current B x [a.u.] B x [a.u.] B z > 0 axial flaw ccw current Axial Position 85
Uniform Field effect of coating thickness on axial magnetic flux density B x (ferrous steel, 5 kHz, δ ≈ 0.25 mm, 30-mm-long solenoid) 8 advantages: slot size 7 • 50 × 5 mm testing through coatings 6 20 × 2 mm • depth information 20 × 1 mm • 5 limited boundary effects Δ B x [%] 4 disadvantages: 3 • reduced sensitivity • sensitivity to geometry 2 • flaw orientation 1 0 0 5 10 15 20 Coating Thickness [mm] Axial Flaw (parallel to B , normal to E ) rate of increase of the minimum of B x with 2-mm-diameter coil, ferrous steel slot depth at the center 8 30 B x at 5 kHz Δ B xm per 1 mm Slot Depth [%] 7 B z at 5 kHz 25 6 B x at 50 kHz Δ B x and Δ B z [%] 20 B z at 50 kHz 5 4 15 3 10 40-mm-long 2 solenoid 5 12-mm-long 1 solenoid 0 0 0 10 20 30 40 0 0.5 1 1.5 2 2.5 Slot Depth [mm] Slot Depth [mm] changes normalized to B x 0 86
Flaw Orientation 0.17 0.025 0.16 0.020 transverse flaw axial flaw (normal to B ) 0.15 0.150 (normal to E ) B x [T] B z [T] 0.14 0.100 0.13 0.05 axial flaw (normal to E ) transverse flaw 0.12 (normal to B ) 0 0.11 -0.05 0 1 2 3 4 5 0 1 2 3 4 5 Scanning time [a. u.] Scanning Time [a. u.] eddy current mode magnetic flux mode Magnetic Flux Mode N I electromagnet magnetometer crack Tangential Magnetic Field Normal Magnetic Field Lateral Position Lateral Position 87
5.2 Direct Current Potential Drop Inductive versus Galvanic Coupling potential magnetic field drop injection V I I current probe coil specimen specimen eddy currents electric current advantages of galvanic coupling dc and low-frequency operation constant coupling (four-point measurement) awkward shapes absolute measurements inherently directional 88
Thin-Plate Approximation 2 a combined electric current and potential field 2 b I (+) V (+) I (-) V (-) t << a V (+) V (-) I (+) I (-) ρ I = ρ = + − E r ( ) J r ( ) Δ = ( ) − ( ) V V V π 2 rt ∞ ∞ ρ I dr [ ] Δ V = 2 V a ( − b ) − V a ( + b ) = = V r ( ) ∫ E r dr ( ) ∫ π 2 t r r r ρ ρ + I I a b V r ( ) = − ln r + const Δ = π V ln π − 2 t t a b Lateral Spread of Current Distribution I = J r ( ) 2 π rt y I J (0,0) = π at J (0, w ) V (+) V (-) 2 I a = J (0, ) w π 2 + 2 2 + 2 2 a w t a w I (+) I (-) x 2 w I a J (0, ) w = J (0,0) π 2 + 2 ( a w ) t J (0,0) = 2 J (0, w ) 2 2 a 2 2 + J (0,0) a w 2 = = 2 2 J (0, w ) a 2 = w a 2 89
Thick-Plate Approximation combined electric current and potential field 2 a 2 b I (+) V (+) V (-) I (-) I (+) V (+) I (-) V (-) ≈ t >> a I ρ = ρ = + − E r ( ) J r ( ) Δ = ( ) − ( ) V V V 2 2 π r ∞ ∞ I ρ dr [ ] Δ = − − + V 2 V a ( b ) V a ( b ) = = V r ( ) ∫ E r dr ( ) ∫ π 2 2 r r r ρ I ρ ⎡ ⎤ I 1 1 = + Δ = − V r ( ) const V ⎢ ⎥ π 2 r π − + ⎣ a b a b ⎦ Finite Plate Thickness 2 a I (+) V (+) I (-) V (-) 2 b I (+) I (-) n = +2 n = +1 ∞ ρ I = ∑ V r ( ) π 2 + 2 1/ 2 2 [ r (2 nt ) ] =−∞ n V (+) V (-) 2 t n = 0 t ρ ∞ ⎡ I 1 Δ V = ∑ ⎢ π − 2 + 2 1/ 2 [( a b ) (2 nt ) ] ⎣ n =−∞ ⎤ 1 − ⎥ 2 2 1/ 2 + + n = -1 [( a b ) (2 nt ) ] ⎦ n = -2 90
Resistance versus Thickness Δ V = = ρΛ R I 2 b + Λ = π 1 a b lim Λ = π lim ln 2 2 − →∞ a b t t a − b → t 0 10 a = 3 b finite thickness Normalized Resistance, Λ thin-plate appr. thick-plate appr. 1 0.1 0.01 0.1 1 10 100 Normalized Thickness, t / a Crack Detection by DCPD intact specimen cracked specimen I (+) V (+) I (-) I (+) V (+) I (-) V (-) V (-) c t ( ) + ( ) − − = Δ + − V V V ( ) ( ) V − V = Δ V 0 c infinite slot 3 Normalized Potential Drop, Δ V c / Δ V 0 a = 3 b 2 a / t = 0.44 1.2 1.8 1 0 0.2 0.4 0.6 0.8 1 Normalized Crack Depth, c / t 91
Technical Implementation of DCPD + power polarity _ supply switch + V s _ electrodes specimen • low resistance, high current • thermoelectric effect, pulsed, alternating polarity • control of penetration via electrode separation • low sensitivity to near-surface layer • no sensitivity to permeability 5.3 Alternating Current Potential Drop 92
Direct versus Alternating Current DCPD ACPD • higher resistance, lower current • no thermoelectric effect • control of penetration via frequency • higher sensitivity to near-surface layer • sensitivity to permeability Thin-Plate/Thin-Skin Approximation 2 a 2 b I (+) V (+) V (-) I (-) t << a Δ ρ + V a b = π lim ln I t a − b → f 0 Δ ρ + ⎧ V ⎫ ≈ a b Re ln ⎨ ⎬ π − ⎩ I ⎭ T a b { } ≈ δ T min t , ρ δ = π μ f ⎧ Δ ⎫ ∝ ρμ + V f a b lim Re ⎨ ⎬ ln ⎩ ⎭ π − →∞ I a b f 93
Skin Effect in Thin Nonmagnetic Plates analytical prediction a = 20 mm, b = 10 mm, t = 2 mm a = 20 mm, b = 10 mm, σ = 50 %IACS 10 3 10 3 1 %IACS 0.05 mm 2 %IACS 0.1 mm Resistance [µ Ω ] Resistance [µ Ω ] 10 2 10 2 5 %IACS 0.2 mm 10 %IACS 0.5 mm 20 %IACS 1 mm 50 %IACS 2 mm 10 1 10 1 100 %IACS 5 mm f t f t 10 0 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 0 10 1 10 2 10 3 10 4 10 5 Frequency [Hz] Frequency [Hz] δ = ≈ ( f f ) t t 1 = f t πμ σ 2 t 0 Skin Effect in Thick Nonmagnetic Plates 304 austenitic stainless steel, σ = 2.5 %IACS, experimental a = 10 mm, b = 7.5 mm 10 4 0.05 mm 0.1 mm 0.2 mm 0.5 mm 10 3 Resistance [µ Ω ] 1 mm 2 mm 2.5 mm 6.25 mm 10 2 10 mm 20 mm 50 mm 10 1 10 0 10 1 10 2 10 3 10 4 10 5 Frequency [Hz] 94
Current Distribution in Ferritic Steel FE predictions (Sposito et al. , 2006) f = 0.1 Hz f = 50 Hz f = 1 kHz a = 10 mm, b = 5 mm, t = 38-mm, c = 10 mm (0.5-mm-wide notches, two separated by 5 mm) Thin-Skin Approximation 2 a 2 a 2 b Δ 2 b V � = = + Z R i X I c ρ + ρ + + ln a b a b 2 c ≈ πδ ≈ πδ R R ln 0 c − − a b a b * * ≈ Γ ≈ Γ R R R R 0 0 c c + + + ln a b ln a b 2 c Γ = Γ = 0 c − − a b a b 2 μρ f * = R Electrode Gain, Γ 0 π R − R Γ − Γ 1 c 0 c 0 = ≈ K c Γ R 0 0 1 2 c 0 ≈ Γ lim K c 1 2 3 + → a b c 0 0 Electrode Shape Factor, a / b 95
Technical Implementation of ACPD V r low-pass filter A/D converter V q 90º phase differential low-pass oscillator shifter driver filter + V s PC _ electrodes processor specimen frequency range: 0.5 Hz - 100 kHz driver current: 10-200 mA resistance range: 1-10,000 µ Ω common mode rejection: 100-160 dB . Application Example: Weld Penetration clamshell 2 d = 0.120” catalytic converter welding current injection edge weld voltage weldment a = 0.160” sensing b = 0.080” d w = 0.054” w electrode separation ( b ) weld penetration ( w ) 200 80 70 Fracture Surface [mils] 150 60 Resistance [µ Ω ] 50 100 40 b = 30 120 mils 50 20 100 mils 10 80 mils 0 0 0 20 40 60 80 100 120 0 10 20 30 40 50 60 70 80 Weld Penetration [mil] NDE [mil] 96
Application Example: Erosion Monitoring ρ ≈ ρ + β − ( ) T [1 ( T T )] 0 0 130 β ≈ 0.001 [1/ºC] 120 Resistivity [µ Ω cm] 110 internal 100 301 erosion/corrosion 302 90 303 304 80 309 310 70 316 321 pipe 60 347 403 50 0 200 400 600 800 Temperature [ºC] before compensation after compensation 25 33.0 25 33.0 24 32.8 24 32.8 Temperature [ºC] Temperature [ºC] Resistance [µ Ω ] Resistance [µ Ω ] 23 32.6 23 32.6 22 32.4 22 32.4 21 32.2 21 32.2 erosion erosion 20 32.0 20 32.0 0 5 10 15 20 0 5 10 15 20 Time [day] Time [day] 97
6 Special Methods 6.1 Microwave Techniques 6.2 Dielectric Measurements 6.3 Thermoelectric Measurements 6.1 Microwave Techniques 98
Electromagnetic Spectrum Frequency [Hz] 4 10 6 10 8 10 10 12 10 14 16 10 18 10 20 10 22 10 24 10 10 10 Wavelength [m] 4 10 2 10 0 -2 10 -4 10 -6 -8 10 -10 10 -12 10 -14 10 -16 10 10 10 typical lattice constant Energy [eV] -10 10 -8 10 -6 -4 10 -2 10 0 2 10 4 10 6 10 8 10 10 10 10 10 γ rays cosmic rays visible microwave light radio frequency IR light UV light X-rays − 34 − 19 E = h ν = eV , h ≈ 6.63 × 10 Js, e ≈ 1.6 × 10 C Electromagnetic Waves Plane waves: E e ω − = = 0 i ( t k x ) H e ω − 0 i ( t k x ) E E e e = = H H e e y y y z z z = − ωμ σ + ωε k i ( i ) ωμ E i η = 0 = σ+ ωε H i 0 in dielectrics: in conductors: ω 1 i k = − k = δ δ c − δ ω − δ E e ω − = x / i ( t x / ) = 0 i ( t x c / ) E E e e e E e 0 y y − δ − ω − δ = x / i ( t x / ) H e ω − H H e e e = 0 i ( t x c / ) H e 0 z z 1 1 c 1 δ = 0 = = = ≈ × 8 c c 3 10 m/s 0 π μσ f μ ε ε μ ε n 0 0 r 0 0 ωμ + i 1 i μ η μ η = = 0 0 0 η = ≈ η = ≈ Ω 377 σ σδ 0 ε ε ε n 0 r 0 99
Reflection/Transmission between Dielectrics y I dielectric II dielectric incident x reflected transmitted η η 0 0 η = η = , I II n n I II − n n = I II R + n n I II • strong penetration • perceivable reflection Reflection from Conductors y I dielectric II conductor incident x transmitted reflected “diffuse” wave 1 δ = ≈ 0 π μσ f ωμ η i 0 η = << η = II I σ n η − η = II I ≈ − R 1 η + η II I • negligible penetration • almost perfect reflection with phase reversal 100
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