Electromagnetic NDE Peter B Nagy Research Centre for

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

Aims and Goals Aims 1 2 The 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. To 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: 1 fundamental physical principles of EM NDE methods 2 operation of basic EM NDE techniques 3 functions of simple EM NDE instruments 4 main applications of EM NDE

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

1 Electromagnetism 1. 1 Fundamentals 1. 2 Electric Circuits 1. 3 Maxwell's Equations 1. 4 Electromagnetic Wave Propagation

1. 1 Fundamentals of Electromagnetism

Electrostatic Force, Coulomb's Law Fe y Q 1 x z r Q 2 Fe Fe Coulomb force Q 1, Q 2 electric charges ( ne, e 1. 602 10 -19 As) er unit vector directed from the source to the target r distance between the charges ε permittivity (ε 0 ≈ 8. 85 10 -12 As/Vm) dρ ρ r x Q 1 Fe independent of x d. Q 2 infinite wall of uniform charge density q

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

Electric Field, Point Sources monopole dipole E 1 +Qs E 2 d +Qs -Qs E 1

Electric Field of Dipole z Ez ER θ r+ +Qs r E P r R d -Qs

Electric Dipole in an Electric Field E +Q Fe Fe pe -Q pe electric dipole moment Q electric charge d distance vector E electric field Fe Coulomb force Te twisting moment or torque

Electric Flux and Gauss’ Law D d. S Qenc closed surface S q charge (volume) density D electric flux density (displacement) E electric field (strength, intensity) ε permittivity electric flux Qenc enclosed charge

Electric Potential dℓ dℓ E B dℓ A Fe Q dℓ W work done by moving the charge Fe Coulomb force ℓ path length E electric field Q charge U electric potential energy of the charge V potential of the electric field

Capacitance C capacitance V voltage difference Q stored charge +Q +Q A +Q E dℓ E E l -Q -Q dℓ -Q

Current, Current Density, and Conductivity d. A E I current Q transferred charge t time J current density A cross section area n number density of free electrons vd mean drift velocity e charge of proton m mass of electron τ collision time Λ free path v thermal velocity k Boltzmann’s constant T absolute temperature σ conductivity

Resistivity, Resistance, and Ohm’s Law I A + V _ dℓ dℓ V voltage I current R resistance P power σ conductivity ρ resistivity L length A cross section area

Magnetic Field Q dv B Fm F Lorenz force v velocity B magnetic flux density Q charge pm magnetic dipole moment B pm +I -I (no magnetic monopole) N number of turns I current A encircled vector area

Magnetic Dipole in a Magnetic Field B Fm pm -I +I Fm pm magnetic dipole moment Q charge v velocity R radius vector B magnetic flux density Fm magnetic force Tm twisting moment or torque

Magnetic Field Due to Currents Coulomb Law: dℓ Biot-Savart Law: H dℓ I r dℓ H magnetic field μ magnetic permeability

Ampère’s Law Gauss’ Law: Ampère’s Law: Biot-Savart Law: infinite straight wire dℓ ℓ r R s I H dℓ Ampère’s Law:

Induction, Faraday’s Law, Inductance B I N V F E induced electric field B magnetic flux density t time Є induced electromotive force s boundary element of the loop Φ magnetic flux S surface area of the loop μ magnetic permeability N number of turns I current Λ geometrical constant L (self-) inductance

Electric Boundary Conditions Gauss' law: Faraday's law: xn xn medium II DII q. II boundary DII, n DI, t DI, n q. II DI q. I EI, t xt EII, n EII, t q. I EI, n EI medium I tangential component of the electric field E is continuous normal component of the electric flux density D is continuous medium I xt

Magnetic Boundary Conditions Gauss' law: Ampère's law: xn xn medium II BII q. II BII, n BI, t boundary BII, t BI, n q. II BI q. I HI, t xt HII, n HII, t q. I HI, n HI medium I tangential component of the magnetic field H is continuous normal component of the magnetic flux density B is continuous medium I xt

1. 2 Electric Circuits

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

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

DC Impedance Matching + _

AC Impedance I V I V

AC Power real notation reminder: complex notation correspondence

AC Impedance Matching

1. 3 Maxwell's Equations

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

Maxwell's Equations Field Equations: Ampère's law: Faraday's law: Gauss' law: Constitutive Equations: conductivity permittivity permeability (ε 0 ≈ 8. 85 10 -12 As/Vm) (µ 0 ≈ 4π 10 -7 Vs/Am)

1. 4 Electromagnetic Wave Propagation

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

Wave Propagation versus Diffusion k wave number c wave speed Propagating wave in free space: Propagating wave in dielectrics: n refractive index Diffusive wave in conductors: δ standard penetration depth

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

Polarization Plane waves propagating in the x-direction: z Ez z E Ey linear polarization z E y elliptical polarization y circular polarization

Reflection at Normal Incidence y I medium II medium incident x reflected Boundary conditions: transmitted

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

Axial Skin Effect y propagating wave diffuse wave x dielectric (air) δ standard penetration depth conductor Normalized Depth Profile, F 1 magnitude real part 0. 8 0. 6 0. 4 0. 2 0 -0. 2 0 1 2 Normalized Depth, x / δ 3

Transverse Skin Effect r current, I current density 2 a z conductor rod Normalized Current Density, J/JDC magnitude, 8 a/δ = 1 a/δ = 3 a/δ = 10 7 6 5 Jn 4 3 2 1 0 0 0. 2 0. 4 0. 6 Normalized Radius, r/a 0. 8 1 nth-order Bessel function of the first kind

Transverse Skin Effect r current density current, I 2 a z conductor rod Normalized Resistance, R/R 0 10 1 0. 01 0. 1 1 Normalized Radius, a/δ 10 100