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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