Electromagnetic NDE

Peter B. Nagy

Research Centre for NDE

Imperial College London

2009

Aims and Goals

Aims

1 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.

2 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 methods2 operation of basic EM NDE techniques3 functions of simple EM NDE instruments4 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

x

z

y

r

Q2

Q1

Fe

Fe

Fe Coulomb force

Q1, Q2 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 (ε0 ≈ 8.85 10-12 As/Vm) 1 2

e 24 xQ dQ xd

rr

F e

2 , 2dQ q dA dA d

1e 3

02xQ q x d

r

eF

2 22 2

, d r rr xdr r x

1e 22

x

r x

Q q x drr

eF

1e 2

xQ q

eF

x

dQ2

Q1

Fe

ρ

dρ

r

infinite wall of uniform charge

density q

independent of x

1 2e 24 r

Q Qr

F e

Electric Field, Plane Electrodes

Qt

Fe

x

z

y

e 2t xQ q

eF

infinite wall of uniform charge density q

2 xq

E e

E+Q -Q

A

QqA

charged parallel plane electrodes Q

xq

E e

e tQF E

e tQF E

Electric Field, Point Sources

e 24s t

rQ Q

r

F e

24s

rQ

r

E e

monopole

+Qs

+Qs

-Qs

1 32sQ dE

r

2 34sQ dE

r

+Qs

-Qs

d

E1

E2

E1

dipole

Electric Field of Dipole

z z R RE E E e e

3/ 2 3/ 22 2 2 2

/ 2 / 24 ( / 2) ( / 2)

sz

Q z d z dEz d R z d R

23 3cos 1

4s

zQ dE

r

3/ 2 3/ 22 2 2 24 ( / 2) ( / 2)

sR

Q R REz d R z d R

33 sin 28

sR

Q dEr

2 2 2r z R cosz r sinR r

R

z

+Qs

-Qs

d

θ

rr+

r

PEz

ER

E

Electric Dipole in an Electric Field

+Q

-Q

pe

Fe

E EEE e

e dQ Qd p d e

e e Q T d F d E

e QF E

pe electric dipole moment

Q electric charge

d distance vector

E electric field

Fe Coulomb force

Te twisting moment or torque

Fee e T p E

Electric Flux and Gauss’ Law

q charge (volume) density

D electric flux density (displacement)

E electric field (strength, intensity)

ε permittivity

electric flux

Qenc enclosed charge

closed surface S

D

dS

Qenc

Sd D S

D E

d d D S

Sd dSS e

encV

q dV Q

q D

Electric Potential

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

E

QFe

d

A

BB A ABU U U W

edW F d

BAB

AW Q Ed

U V Q

BB A

AV V V Ed

Capacitance

+Q

-Q

E

C capacitance

V voltage difference

Q stored charge

Q CV

+

-

S+ -

SV V V Ed

QCV

E

+Q

-Q

A

-Q

E

+Q

d

QDA

ACDE

V E

Current, Current Density, and Conductivity

I currentQ transferred charget timeJ current densityA cross section arean number density of free electrons

vd mean drift velocity

e charge of protonm mass of electronτ collision timeΛ free pathv thermal velocityk Boltzmann’s constantT absolute temperatureσ conductivity

dQIdt

dI d J A

I dJ A

dne J v

ddQ ne d dt v A

dm e v E

v

21 32 2

mv kT

E

dA

2nem

J E E

Resistivity, Resistance, and Ohm’s Law

V voltage

I current

R resistance

P power

σ conductivity

ρ resistivity

L length

A cross section area

I

+_V

A

d

0 0

L Ld dRA A

i i

i

LRA

1

LRA

+

-

S+ -

SV V V Ed

0 0

L LJ dV d I I RA

VRI

dU dQP V V Idt dt

Magnetic Field

BQ

Fm

dv

e QF E

m Q F v B

( )Q F E v B

F Lorenz forcev velocityB magnetic flux densityQ charge

+I -I

B

pm magnetic dipole moment

(no magnetic monopole)N number of turnsI currentA encircled vector area

m N Ip A

pm

Magnetic Dipole in a Magnetic Field

m Q F v B

pm magnetic dipole moment

Q charge

v velocity

R radius vector

B magnetic flux density

Fm magnetic force

Tm twisting moment or torque

m N Ip A

+I

-Ipm

Fm

B

Fm

2m 2 r v

Qv RR

p e e

2A RQN I

2 Rv

m12

Q p R v

m m12

T R F

22

m m m0

1 1cos2 2

T R F d R F

m m T p B

Magnetic Field Due to Currents

2 34 4s s

rQ Q

r r

E e rCoulomb Law:

D E

B H

Biot-Savart Law: 2 34 4rI d Id

r r

H e e r

d

d

I

dℓ r

HH magnetic field

μ magnetic permeability

24 rI d

r

H e e

Ampère’s Law

24 rI dd

r

H e e

encS

d QD SGauss’ Law:

infinite straight wire

2 2 2 3/ 244 ( )I d R I R dd

rr R

H e e

2 2 3/ 202 2( )

I R d IHRR

2H ds H R I

d

I

dℓ

R

Hrℓ

s

2IH

R

Biot-Savart Law:

Ampère’s Law:

Ampère’s Law: encd IH s

H J

N I

Є dV Ndt

Induction, Faraday’s Law, Inductance

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

I N

V

μ magnetic permeability

N number of turns

I current

Λ geometrical constant

L (self-) inductanceI LN

2L N

Sd B S

Є ddt

t

BE

dIV Ldt

B

Є d E s

ЄS

dt

B S

Electric Boundary ConditionsFaraday's law:

t

BE

Gauss' law:

q D

xt

medium I

medium II

DI

boundary

DII

DII,t

DII,n

DI,n

DI,t

xn

xt

medium I

medium II

EI

EIIEI,t EII,n

EI,n

EII,t

xn

I,n II,nD D

I I,n II II,nE E

I,t II,tE E

I I,n II II,ntan tanE E I II

I II

tan tan

tangential component of the electric field E is continuousnormal component of the electric flux density D is continuous

Magnetic Boundary ConditionsAmpère's law:

t

DH J

Gauss' law:

0 B

xt

medium I

medium II

BI

boundary

BII

BII,t

BII,n

BI,n

BI,t

xn

xt

medium I

medium II

HI

HII

HII,t

HII,n

HI,n

HI,t

xn

I,n II,nB B

I I,n II II,nH H

I,t II,tH H

I I,n II II,ntan tanH H I II

I II

tan tan

tangential component of the magnetic field H is continuousnormal component of the magnetic flux density B is

continuous

1.2 Electric Circuits

Є

Electric Circuits, Kirchhoff’s Laws

Є electromotive force

Vi potential drop on ith element

Kirchhoff’s junction rule (current law):

Kirchhoff’s loop rule (voltage law):

0iV

0E dI

+_

1R 2R

4R

3R1V 2V

4V3V0V

0iI

encS

Q d D S

Ii current flowing into a junction from the ith branch

+_Є

1I 2I

4I

1R 2R

4R

3R

Circuit Analysis

Loop Currents:

Kirchhoff’s Laws:

+_Є

1I 2I

4I

1R 2R

3R1V 2V

3V0V

+_Є

1I 2I

4I

1R 2R

4R

3R1i 2i

4R

4V1 2 4

1 2 40V V V

R R R

1 4 0 0V V V

2 3 4 0V V V

32

2 30VV

R R

1 1 1 2 4 0( ) 0i R i i R V

2 2 2 3 1 2 4( ) 0i R i R i i R

DC Impedance Matching

2g

2g g, where

(1 )

V RP

R R

22 V

P I V I RR

g g

g gand

V V RI V

R R R R

2g

3g

1(1 )

VdPd R

2g

max gg

when4V

P R RR

_ VgV

gR

R+

P IVW QV

AC ImpedanceI

VdIV Ldt

I

V1V I dtC

I

V V R I

VZ i LI

VZ RI

1VZI i C

0

0ZiVZ R i X Z e

I

0 2 20

VZ R XI

-1arg( ) - tanZ V IXZR

( )0 0( ) Ii t i tI t I e I e

( )0 0( ) Vi t i tV t V e V e

ReI I

ReV V

0 0 IiI I e

0 0 ViV V e

AC Power

ReI I ( )0 0( ) Ii t i tI t I e I e 0( ) cos( )II t I t

ReV V ( )0 0( ) Vi t i tV t V e V e 0( ) cos( )VV t V t

* *0 0

1 1( ) ( )2 2

P I t V t I V ( ) ( )P I t V t ReP P

( )0 0

12

I ViP I V e 0 01 cos( )2 I VP I V

real notation complex notation correspondence

cos( ) cos cos sin sin

cos( ) cos cos sin sin

1 1cos( ) cos( ) cos cos2 2

cos sinie i

reminder:

AC Impedance Matching

VgV

gZ

Z

ReP P

2 *g*

*g g

1 Re Re2 2 ( )( )

V ZP I VZ Z Z Z

*g g g,Z Z R R X X

2g

maxg8

VP

R

2g

2Re2 4

g g

g

V R i XP

R

1.3 Maxwell's Equations

Vector Operations

0

limSS S

AA e dℓCurl of a vector:

0lim yS x z

V

dAA A

V x y z

A SA

Divergence of a vector:

x y zx y z

e e eGradient of a scalar:

2 2 22

2 2 2x y z

Laplacian of a scalar:

2 2 2 2x x y y z zA A A A e e eLaplacian of a vector:

2( ) ( ) A A AVector identity:

x y zx y z

e e eNabla operator:

2 2 22

2 2 2x y z

Laplacian operator:

y yx xz zx y z

A AA AA Ay z z x x y

A e e ea

Maxwell's Equations

Ampère's law:

Faraday's law:

Gauss' law:

Gauss' law:

t

DH J

t

BE

q D

0 B

Field Equations:

conductivity J E

permittivity D E

permeability

B H

Constitutive Equations:

(ε0 ≈ 8.85 10-12 As/Vm)

(µ0 ≈ 4π 10-7 Vs/Am)0 r

0 r

1.4 Electromagnetic Wave Propagation

Electromagnetic Wave Equation

Maxwell's equations:

( )it

DH J E

it

BE H

0 E

0 H

( ) ( )i i H H

( ) ( )i i E E

2( ) ( ) A A A

2 ( )i i E E

2 ( )i i H H

2 ( )k i i

2 2( )k E 0

2 2( )k H 0

( )0

i t k xy y yE E e E e e

( )0

i t k xz z zH H e H e e

Example plane wave solution:

Wave equations:

Harmonic time-dependence: 0 0andi t i te e E E H H

Wave Propagation versus Diffusion

Propagating wave in free space:

/ ( / )0

x i t xyE e e E e

/ ( / )0

x i t xzH e e H e

Diffusive wave in conductors:

kc

0 0

1c

1 ik i

1f

( / )0

i t x cyE e E e

( / )0

i t x czH e H e

2 ( )k i i

δ standard penetration depth

c wave speed

k wave number

Propagating wave in dielectrics:

d0 0 r

1c r

d

cnc

n refractive index

Intrinsic Wave Impedance( )

0i t k x

y y yE E e E e e ( )0

i t k xz z zH H e H e e

( )it

DH J E

( )0

z i t k xy y

H i k H ex

H e e

( )k i i

Propagating wave in free space:0

00

377

Propagating wave in dielectrics:0 0

0 r n

Diffusive wave in conductors:1i i

0

0

E iH i

PolarizationPlane waves propagating in the x-direction:

( ) ( )0 0

i t k x i t k xy y z z y y z zE E E e E e E e e e e

( ) ( )0 0

i t k x i t k xz z y y z z y yH H H e H e H e e e e

0 00

0 0

y z

z y

E EH H

0 0 0 0y zi iy y z zE E e E E e

y

z

y

z

y

z

Ey

EzE

0º (or 180º)y z

linear polarization elliptical polarization

90º (or 270º)y z

circular polarization

E E

Reflection at Normal Incidence

x

y

incident

reflected transmitted

I( )i i0

i t k xyE e E e

Ii0 ( )i

Ii t k x

zE e

H e

I( )r r0

i t k xyE e E e

Ir0 ( )r

Ii t k x

zE e

H e

II( )t t0

i t k xyE e E e

IIt0 ( )t

IIi t k x

zE e

H e

I medium II medium

Boundary conditions:

( 0 ) ( 0 )y yE x E x i0 r0 t0E E E

( 0 ) ( 0 )z zH x H x i0 r0 t0H H H

i0 r0 t0

I I II

E E E

r0 II I

i0 II I

ERE

t0 II

i0 II I

2ETE

Reflection from Conductors

x

y

incident

reflected transmitted“diffuse” wave

I dielectric II conductor

1 0f

0II I

in

II I

II I1R

negligible penetration

almost perfect reflection with phase reversal

Axial Skin Effect

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3Normalized Depth, x / δ

Nor

mal

ized

Dep

th P

rofil

e, F

magnitude real part

0 ( ) i tyE F x e E e

0 ( ) i tzH F x e H e

δ standard penetration depth

/ /( ) x i xF x e e

1f

x

y

propagating wave diffuse wave

dielectric (air) conductor

Transverse Skin Effect

0 0( )zE E J k r

1f

2k i 1 ik

012 ( )

k IEa J k a

Jn nth-order Bessel function

of the first kind

02 1

( )( )2 ( )z

k a J k rIJ rJ k aa

z

r

current density

conductor rod

current, I 2a

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1Normalized Radius, r/a

Nor

mal

ized

Cur

rent

Den

sity

, J/J

DC

a/δ = 1a/δ = 3a/δ = 10

magnitude, DC 2IJa

Transverse Skin Effect

z

r

current density

conductor rod

current, I 2a

0.1

1

10

100

0.01 0.1 1 10 100

Normalized Radius, a/δ

Nor

mal

ized

Res

ista

nce,

R/R

0

R

0R R

VZ R i XI

0 2RA a

0

1

( )( )2 ( )

JGJ

0 ( )Z R G k a

/lim (1 )

2a

aG i

/lim

2aR

a