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
2
3
4
fundamental physical principles of EM NDE methods
operation of basic EM NDE techniques
functions of simple EM NDE instruments
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 
Fe
y
Q1
x
z
r
Q2
Fe
Q1 Q2
er
4  r 2
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)
dFe 
Q1 dQ2 x
ex
4  r 2 r
dQ2  q dA, dA  2  d 
Q1 q x e x   d 
Fe 

2    0 r3
dρ
ρ
r
x
Q1
Fe
independent of x
dQ2
infinite wall of
uniform charge
density q
d
  r 2  x2 ,

dr
Fe 
r
r 2  x2
Q1 q x e x  dr
 2
2
rxr
Fe 
Q1 q e x
2

r

Electric Field, Plane Electrodes
Fe  E Qt
y
infinite wall of uniform
charge density q
Qt
x
Fe
z
Fe 
E 
Qt q e x
2
q
ex
2
charged parallel plane
electrodes Q
A
+Q
E
-Q
E 
q 
l
q
ex

Q
A
Electric Field, Point Sources
Fe  E Qt
monopole
dipole
E1
+Qs
+Qs
E2
d
+Qs
-Qs
-Qs
E1
Fe 
Qs Qt
er
4  r 2
E1 
Qs d
2  r 3
E 
Qs
er
4  r 2
E2 
Qs d
4  r 3
Electric Field of Dipole
z
Ez
ER
θ
r+
E
P
r
+Qs
r
R
d
-Qs
E  E z e z  ER e R
r 2  z 2  R2
z  r cos 


Qs 
z  d /2
z  d /2

Ez 

4   ( z  d / 2)2  R 2 3 / 2 ( z  d / 2)2  R 2 3 / 2 
 






Qs 
R
R

ER 


4  ( z  d / 2)2  R 2 3 / 2 ( z  d / 2)2  R 2 3 / 2 
 




R  r sin 
Ez 


Qs d
3cos2   1
3
4  r
ER 
3Qs d
sin 2
8  r 3
Electric Dipole in an Electric Field
pe  Q d  Q d ed
E  E eE
E
Fe  E Q
+Q
Te  d  Fe  d  E Q
Fe
Fe
Te  pe  E
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 q
D  E
dS  e S dS
D
d   D dS
dS
   D dS
S
   q dV  Qenc
Qenc
V
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
dW   Fe d
B
WAB   Q  E d
A
U  U B  U A  WAB
E
B
d
A
U  VQ
Fe
B
V  VB  VA    E d
Q
A
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
Q  CV
C
capacitance
V
voltage difference
Q
stored charge
+Q
+Q
A
+Q
E
d
E
E
l
-Q
-Q
-Q
S+
V  V+  V-    E d
S-
C 
Q
V
D 
Q
A
D
E 

V  E








C 
A
Current, Current Density, and Conductivity
dA
E
dQ
dt
I
current
Q
transferred charge
I   J dA
t
time
dI  J dA
J
current density
dQ   ne v d dA dt
A
cross section area
n
number density of free electrons
vd
mean drift velocity
m vd
 eE


 
v
e
charge of proton
m
mass of electron
τ
collision time
Λ
free path
1 2 3
mv  k T
2
2
v
thermal velocity
k
Boltzmann’s constant
T
absolute temperature
σ
conductivity
I 
J   ne v d
J 
ne2 
E  E
m
Resistivity, Resistance, and Ohm’s Law
I
A
+
V _
d
S+
V  V+  V-    E d
SLJ
V  
0
L d
d  I
0A
 IR
V
R 
I
dU
dQ
P 
V
VI
dt
dt
V
voltage
I
current
R
resistance
P
power
σ
conductivity
ρ
resistivity
L
length
A
cross section area
L d
R  
0A
 
L d
 
0 A
1

 L
R   i i
Ai
R 
L
A
Magnetic Field
Fe  Q E
Fm  Q v  B
Q
dv
B
Fm
F  Q (E  v  B)
F
Lorenz force
v
velocity
B
magnetic flux density
Q
charge
B
pm  N I A
pm
+I
pm
-I
magnetic dipole moment
(no magnetic monopole)
N
number of turns
I
current
A
encircled vector area
Magnetic Dipole in a Magnetic Field
pm  N I A
pm 
B
Qv
 R 2 e r  ev
2 R
pm 
Fm
pm
Fm  Q v  B
-I
Tm 
+I
Fm
Q
NI 

2 R
 
v
1
Q R v
2
1
R  Fm
2
Tm  p m  B
A   R2
1 2 2
1
Tm 
 cos  R Fmd   R Fm
2 0
2
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
E 
Coulomb Law:
Qs
Qs
er  
r
4  r 2
4  r 3
D  E
Biot-Savart Law:
dH 
Id
I
e  er 
d r
4r2
4  r3
H  
Id
e  er
4r2
B  H
H
dℓ
I
r
d
H
magnetic field
μ
magnetic permeability
Ampère’s Law
 D dS  Qenc
Gauss’ Law:
S
 H  J
Ampère’s Law:
 H ds  I enc
dH 
Biot-Savart Law:
infinite straight wire
dH 
dℓ
r
ℓ
R
s
Id
e  er
4r2
Id R
IR
d
e 
e
4  ( 2  R 2 )3 / 2
4r2 r
H
H 
d
IR 
d
I

 2
2  0 (  R 2 )3 / 2
2 R
Ampère’s Law:
I
 H  ds  H  2  R  I
H 
I
2 R
Induction, Faraday’s Law, Inductance
 E  
B
B
t

Є    B dS
t S
   B dS
S
Є   E ds
d
Є  
dt
V  Є  N
I
N
V
dI
dt
L  N2 

 
IL
N
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
d
dt
  N I 
V  L
E
Electric Boundary Conditions
Gauss' law:
Faraday's law:
 D q
 E  
xn
B
t
xn
medium II
medium II
DII
II
II
DII,n
DI,t
boundary
DII,t
DI,n
EII
EI,t
xt
DI I
EII,n
EII,t
I
EI,n
EI
medium I
DI,n  DII,n
I EI,n  II EII,n
medium I
tan I
tan II

I
II
EI,t  EII,t
tan I EI,n  tan II EII,n
tangential component of the electric field E is continuous
normal component of the electric flux density D is continuous
xt
Magnetic Boundary Conditions
Gauss' law:
Ampère's law:
 B  0
 H  J 
xn
D
t
xn
medium II
medium II
BII
II
II
BII,n
BI,t
boundary
BII,t
BI,n
HII
HI,t
xt
BI I
HII,n
HII,t
I
HI,n
HI
medium I
BI,n  BII,n
I H I,n  II H II,n
medium I
tan I
tan II

I
 II
H I,t  H II,t
tan I H I,n  tan II H II,n
tangential component of the magnetic field H is continuous
normal component of the magnetic flux density B is continuous
xt
1.2 Electric Circuits
Electric Circuits, Kirchhoff’s Laws
Kirchhoff’s loop rule (voltage law):
R1
+
Є _
V0
E d 0
R2
I
V2
V1
V4
V3
 Vi  0
R3
R4
Є
electromotive force
Vi
potential drop on ith element
Kirchhoff’s junction rule (current law):
R1
Є
I2
I1
Qenc   D dS
R2
S
I4
+
_
 Ii  0
R3
R4
Ii
current flowing into a
junction from the ith branch
Circuit Analysis
Kirchhoff’s Laws:
V1  V4  V0  0
R1
Є
+
_
V0
V1
R2
I2
I1
I4
V4
V2
V2  V3  V4  0
V3
R3
V1 V2 V4


 0
R1 R2 R4
V2 V3

 0
R2 R3
R4
Loop Currents:
R1
+
Є _
I2
I1
R2
i1 R1  (i1  i2 ) R4  V0  0
I4
i2
i1
R4
R3
i2 R2  i2 R3  (i1  i2 ) R4  0
DC Impedance Matching
Rg
Vg
+
_
V
R
W  QV
P  IV
V2
P  I V  I2 R 
I 
P 
Vg
Rg  R
Vg2
V 
and

,
Rg (1  )2
R
Vg R
Rg  R
 
where
R
Rg
Vg2 1  
dP

d
Rg (1  )3
Pmax 
Vg2
4 Rg
when
R  Rg
AC Impedance
I
I
V  L
V
Z 
dI
dt
I
V  RI
V
V
 iL
I
Z 
V 
V
V
 R
I
Z 
V
1

I
i C

V (t )  V0 ei (t  V )  V0 eit
V0  V0 ei V
V  Re V
I (t )  I 0 ei (t  I )  I 0 eit
I 0  I 0 ei I
I  Re I

V
Z  0  R  i X  Z eiZ
I0
V
Z  0 
I0
R2  X 2
arg(Z )  Z  V - I  tan -1
X
R
1
 I dt
C
AC Power
complex notation
correspondence
I (t )  I 0 cos(t   I )
I (t )  I 0 ei (t  I )  I 0 eit
I  Re I
V (t )  V0 cos(t  V )
V (t )  V0 ei (t  V )  V0 eit
V  Re V
real notation
P  I (t )V (t )
P 
1
I 0 V0 cos(I  V )
2
P 
1
1
I (t )V * (t )  I 0 V0*
2
2
P 
1
I0 V0 ei (I  V )
2
reminder:
cos(  )  cos  cos   sin  sin 
cos(  )  cos  cos   sin  sin 
1
1
cos(  )  cos(  )  cos  cos 
2
2
ei  cos   i sin 



P  Re P
AC Impedance Matching
Zg
Vg

V
Z

P  Re P


Vg2
1
Z*
*
P  Re I V

Re 
*
2
2
(
Z

Z
)(
Z

Z
)
g

 g


Zg  Z *
P 
 Rg
 R , Xg   X
 Rg  i X g 
Re 

2
2
 4 Rg 
Vg2
Pmax 
Vg2
8 Rg

1.3 Maxwell's Equations
Vector Operations



 ey
 ez
x
y
z
Nabla operator:
  ex
Laplacian operator:
2    
Gradient of a scalar:
  e x
Curl of a vector:
a
  A 
2
2
2


x 2 y 2 z 2



 ey
 ez
x
y
z
 A dℓ 
e S  lim  

S 0 S 
Ay 
 A
 Ay Ax 
 Ax Az 
 A  e x  z 


  ey 

  ez 
z 
x 
y 
 z
 y
 x
Divergence of a vector:
  A dS 

 Ax Ay Az
 A  lim  S


 

x

y
z
V 0 V



Laplacian of a scalar:
 2  2  2
2
  


x 2 y 2 z 2
Laplacian of a vector:
 2 A   2 Ax e x   2 Ay e y   2 Az e z
Vector identity:
 ( A )  (  A )  2 A
Maxwell's Equations
Field Equations:
Ampère's law:
Faraday's law:
D
t
B
 E  
t
 H  J 
Gauss' law:
 D q
Gauss' law:
 B  0
Constitutive Equations:
conductivity
J  E
permittivity
D  E
permeability
B  H
  0  r
(ε0 ≈ 8.85  10-12 As/Vm)
  0  r
(µ0 ≈ 4π  10-7 Vs/Am)
1.4 Electromagnetic
Wave Propagation
Electromagnetic Wave Equation
E  E0 eit
Harmonic time-dependence:
and H  H 0 eit
Maxwell's equations:
 E  
 H  J 
B
  i  H
t
 ( E)   i  (  i ) E
D
 ( i )E
t
 ( H)   i  (  i ) H
 ( A )  (  A )  2 A
 2E  i  (  i ) E
 E  0
 H  0
Wave equations:
 2H  i  (  i ) H
( 2  k 2 ) E  0
( 2  k 2 ) H  0
k 2   i  (  i )
Example plane wave solution:
E  E y e y  E0 ei (t  k x ) e y
H  H z e z  H 0 ei (t  k x ) e z
Wave Propagation versus Diffusion
k 2   i  (  i )
k
Propagating wave in free space:
wave number

c
k
E  E0 ei(t  x / c ) e y
1
 0 0
c 
c
H  H 0 ei(t  x / c ) e z
wave speed
Propagating wave in dielectrics:
1
0 0  r
cd 
n
n 
c

cd
r
refractive index
Diffusive wave in conductors:
k
 i  
 
1
i



1
 f 
δ
E  E0 e x /  ei (t  x / ) e y
H  H 0 e x /  ei (t  x / ) e z
standard penetration depth
Intrinsic Wave Impedance
E  E y e y  E0 ei (t  k x ) e y
H  H z e z  H 0 ei (t  k x ) e z
k
 i  (  i )
 H  J 
H  
D
 ( i )E
t
H z
e y  i k H 0 ei (t  k x) e y
x
 
E0

H0
i 
  i 
Propagating wave in free space:
0 
0
 377 
0
Propagating wave in dielectrics:
 
0

 0
0  r
n

i 
1 i



Diffusive wave in conductors:
Polarization
Plane waves propagating in the x-direction:
E  E y e y  Ez e z  E y 0 ei (t  k x) e y  Ez 0 ei (t  k x) e z
H  H z e z  H y e y  H z 0 ei (t  k x) e z  H y 0 ei (t  k x) e y
0 
E y0
H z0
E y 0  E y 0 ei y
z
Ez
 
Ez 0
H y0
E z 0  E z 0 e i z
z
E
Ey
linear polarization
 y   z  0º (or 180º)
z
E
y
E
y
elliptical polarization
y
circular polarization
 y   z  90º (or 270º)
Reflection at Normal Incidence
y
I medium
II medium
incident
x
reflected
Ei  Ei0 ei (t  kI x ) e y
Hi 
Ei0 i (t  k x )
I
e
ez
I
transmitted
Er  Er0 ei (t  kI x ) e y
E
H r   r0 ei (t  kI x) e z
I
E t  Et0 ei (t  kII x ) e y
Ht 
Et0 i (t  k x)
II
e
ez
II
Boundary conditions:
E y ( x  0 )  E y ( x  0 )
Ei0  Er0  Et0
H z ( x  0 )  H z ( x  0 )
H i0  H r0  H t0
Ei0 Er0
E

 t0
I
I
II
R 
Er0
  I
 II
Ei0
II  I
T 
Et0
2 II

Ei0
II  I
Reflection from Conductors
y
I dielectric
II conductor
incident
x
reflected
 
II 
transmitted
“diffuse” wave
1
 0
 f 

i 
 I  0

n
R 
II  I
 1
II  I

negligible penetration

almost perfect reflection with phase reversal
Axial Skin Effect
y
propagating wave
E  E0 F ( x) eit e y
H  H 0 F ( x) eit e z
diffuse wave
F ( x )  e  x /  e i x / 
x
dielectric (air)
δ
standard penetration depth
1
 f 
 
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
E z  E0 J 0 (k r )
2a
z
conductor rod
Normalized Current Density, J/JDC
magnitude, J DC 
k 2   i 
 
I
 a2
8
7
E0 
a/δ = 1
a/δ = 3
a/δ = 10
6
5
Jn
k 
1
i



1
 f 
kI
2  a J1(k a)
nth-order Bessel function
of the first kind
4
3
J z (r ) 
2
1
0
0
0.2
0.4
0.6
Normalized Radius, r/a
0.8
1
I k a J 0 (k r )
 a 2 2 J1(k a)
Transverse Skin Effect
r
Z 
current density
current, I
2a
V
 R iX
I
z
conductor rod
Z  R0 G (k a )
R0  

A
 a 2
Normalized Resistance, R/R0
100
G ( ) 
10
R 
1

R  R0
 J 0 ( )
2 J1()
lim G 
a /  
lim R 
0.1
0.01
a /  
0.1
1
Normalized Radius, a/δ
10
100
a
(1  i)
2
2a