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Electromagnetic NDE Peter B. Nagy Research Centre for NDE Imperial College London 2009

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Aims and Goals Aims 1The main aim of this course is to familiarize the students with Electromagnetic (EM) Nondestructive Evaluation (NDE) and to integrate the obtained specialized knowledge into their broader understanding of NDE principles. 2To enable the students to judge the applicability, advantages, disadvantages, and technical limitations of EM techniques when faced with NDE challenges. Objectives At the end of the course, students should be able to understand the: 1fundamental physical principles of EM NDE methods 2operation of basic EM NDE techniques 3functions of simple EM NDE instruments 4main applications of EM NDE

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Syllabus 1Fundamentals of electromagnetism. Maxwell's equations. Electromagnetic wave propagation in dielectrics and conductors. Eddy current and skin effect. 2Electric circuit theory. Impedance measurements, bridge techniques. Impedance diagrams. Test coil impedance functions. Field distributions. 3Eddy current NDE techniques. Instrumentation. Applications; conductivity, permeability, and thickness measurement, flaw detection. 4Magnetic measurements. Materials characterization, permeability, remanence, coercivity, Barkhausen noise. Flaw detection, flux leakage testing. 5Alternating current field measurement. Alternating and direct current potential drop techniques. 6Microwave techniques. Dielectric measurements. Thermoelectric measurements. 7Electromagnetic generation and detection of ultrasonic waves, electromagnetic acoustic transducers (EMATs).

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1 Electromagnetism 1.1Fundamentals 1.2Electric Circuits 1.3Maxwell's Equations 1.4Electromagnetic Wave Propagation

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1.1 Fundamentals of Electromagnetism

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Electrostatic Force, Coulomb's Law x z y r Q2Q2 Q1Q1 FeFe FeFe F e Coulomb force Q 1, Q 2 electric charges ( ne, e 1.602 10 -19 As) e r unit vector directed from the source to the target rdistance between the charges εpermittivity (ε 0 ≈ 8.85 10 -12 As/Vm) x dQ 2 Q1Q1 FeFe ρ dρdρ r infinite wall of uniform charge density q independent of x

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Electric Field, Plane Electrodes QtQt FeFe x z y infinite wall of uniform charge density q E +Q+Q-Q-Q A charged parallel plane electrodes Q

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Electric Field, Point Sources monopole +Qs+Qs +Qs+Qs -Qs-Qs +Qs+Qs -Qs-Qs d E1E1 E2E2 E1E1 dipole

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Electric Field of Dipole R z +Qs+Qs -Qs-Qs d θ r r+r+ r P EzEz ERER E

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Electric Dipole in an Electric Field +Q+Q -Q-Q pepe FeFe E p e electric dipole moment Qelectric charge ddistance vector Eelectric field F e Coulomb force T e twisting moment or torque FeFe

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Electric Flux and Gauss’ Law qcharge (volume) density Delectric flux density (displacement) Eelectric field (strength, intensity) εpermittivity electric flux Q enc enclosed charge closed surface S D dSdS Q enc

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Electric Potential Wwork done by moving the charge F e Coulomb force ℓ path length Eelectric field Qcharge Uelectric potential energy of the charge Vpotential of the electric field E Q FeFe dℓdℓ A B dℓdℓ dℓdℓ dℓdℓ

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Capacitance +Q+Q -Q-Q E Ccapacitance Vvoltage difference Qstored charge dℓdℓ E +Q+Q -Q-Q A -Q-Q E +Q+Q dℓdℓ

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Current, Current Density, and Conductivity Icurrent Qtransferred charge ttime Jcurrent density Across section area nnumber density of free electrons v d mean drift velocity echarge of proton mmass of electron τcollision time Λfree path vthermal velocity kBoltzmann’s constant Tabsolute temperature σconductivity E dAdA

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Resistivity, Resistance, and Ohm’s Law Vvoltage Icurrent Rresistance Ppower σconductivity ρresistivity Llength Across section area I + _ V A dℓdℓ dℓdℓ

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Magnetic Field B Q FmFm dvdv FLorenz force vvelocity Bmagnetic flux density Qcharge +I+I -I-I B p m magnetic dipole moment (no magnetic monopole) Nnumber of turns Icurrent Aencircled vector area pmpm

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Magnetic Dipole in a Magnetic Field p m magnetic dipole moment Qcharge vvelocity Rradius vector Bmagnetic flux density F m magnetic force T m twisting moment or torque +I+I -I-I pmpm FmFm B FmFm

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Magnetic Field Due to Currents Coulomb Law:Biot-Savart Law: dℓdℓ dℓdℓ I dℓdℓ r H Hmagnetic field μmagnetic permeability

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Ampère’s Law Gauss’ Law: infinite straight wire dℓdℓ I dℓdℓ R Hr ℓ s Biot-Savart Law: Ampère’s Law:

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Induction, Faraday’s Law, Inductance Einduced electric field Bmagnetic flux density ttime Єinduced electromotive force sboundary element of the loop Φmagnetic flux Ssurface area of the loop I N V μmagnetic permeability Nnumber of turns Icurrent Λgeometrical constant L(self-) inductance B

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Electric Boundary Conditions Faraday's law:Gauss' law: xtxt medium I medium II DIDI boundary D II D II,t D II,n D I,n D I,t xnxn xtxt medium I medium II EIEI E II E I,t E II,n E I,n E II,t xnxn tangential component of the electric field E is continuous normal component of the electric flux density D is continuous

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Magnetic Boundary Conditions Ampère's law:Gauss' law: xtxt medium I medium II BIBI boundary B II B II,t B II,n B I,n B I,t xnxn xtxt medium I medium II HIHI H II H II,t H II,n H I,n H I,t xnxn tangential component of the magnetic field H is continuous normal component of the magnetic flux density B is continuous

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1.2 Electric Circuits

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Є Electric Circuits, Kirchhoff’s Laws Єelectromotive force V i potential drop on ith element Kirchhoff’s junction rule (current law): Kirchhoff’s loop rule (voltage law): dℓdℓ I + _ I i current flowing into a junction from the ith branch + _ Є

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Circuit Analysis Loop Currents: Kirchhoff’s Laws: + _ Є + _ Є

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DC Impedance Matching _ +

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AC Impedance I V I V I V

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AC Power real notation complex notationcorrespondence reminder:

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AC Impedance Matching

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1.3 Maxwell's Equations

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Vector Operations dℓdℓ Curl of a vector: Divergence of a vector: Gradient of a scalar: Laplacian of a scalar: Laplacian of a vector: Vector identity: Nabla operator: Laplacian operator: a

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Maxwell's Equations Ampère's law: Faraday's law: Gauss' law: Field Equations: conductivity permittivity permeability Constitutive Equations: (ε 0 ≈ 8.85 10 -12 As/Vm) (µ 0 ≈ 4π 10 -7 Vs/Am)

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1.4 Electromagnetic Wave Propagation

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Electromagnetic Wave Equation Maxwell's equations: Example plane wave solution: Wave equations: Harmonic time-dependence:

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Wave Propagation versus Diffusion Propagating wave in free space: Diffusive wave in conductors: δstandard penetration depth cwave speed kwave number Propagating wave in dielectrics: nrefractive index

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Intrinsic Wave Impedance Propagating wave in free space: Propagating wave in dielectrics: Diffusive wave in conductors:

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Polarization Plane waves propagating in the x-direction: y z y z y z EyEy EzEz E linear polarization elliptical polarizationcircular polarization EE

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Reflection at Normal Incidence x y incident reflectedtransmitted I mediumII medium Boundary conditions:

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Reflection from Conductors x y incident reflected transmitted “diffuse” wave I dielectricII conductor negligible penetration almost perfect reflection with phase reversal

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Axial Skin Effect δstandard penetration depth x y propagating wavediffuse wave dielectric (air)conductor

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Transverse Skin Effect J n nth-order Bessel function of the first kind z r current density conductor rod current, I 2a2a magnitude,

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Transverse Skin Effect z r current density conductor rod current, I 2a2a 0.1 1 10 100 0.010.1110100 Normalized Radius, a/δ Normalized Resistance, R/R 0

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