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International Journal of Applied Electromagnetics and Mechanics 18 (2003) 1–5 1
IOS Press
Development of a 3D electromagnetic model
for eddy current tubing inspection:
application to the simulation of probe
eccentricity
Denis Prémel
a
Grégoire Pichenot
a
and Thierry Sollier
a
a
C.E.A Saclay,Batiment 611,91191 Gif-sur-Yvette,France
Tel.:(33) + 01 69 08 59 63;Faz:(33) + 01 69 08 75 97;
E-mail:denis.premel@cea.fr
Abstract.A theoretical and numerical model is developed for the computation of Eddy Current (EC) signals provided by
a differential type probe used for tubing inspection.The main application of interest is to make simulated experiments for
studying the perturbations affecting the EC signal such as the variations of the eccentricity and the probe wobble.3D flaws
are introduced by a variation in the local distribution of the conductivity.The forward problem is handled by using a dyadic
Green’s function approach.Typical impedance plane trajectories obtained with a differential probe are given.
Keywords:non destructive evaluation,forward problem,probe eccentricity
1.Introduction
The inspection of heat exchanger tubes is often carried out by using eddy current nondestructive test-
ing.This technique is based on the analysis of changes in the impedance in one or more coils placed
inside the tube.The EC signal is used to characterize flaws or anomalies.Industrial methods often use
bobbin coils for tubing inspection.This paper describes the progress in developing a 3D computer code
based on the volume integral equations which has the capability to predict quickly the response of an
eddy current probe to 3D flaws.Many works have been done on the development of axisymmetric mod-
els in order to predict EC signals for circumferential flaws [1,2].Others works,based on the Green’s
dyadic formalism,[3,4] are focused on 3Dvolumetric flaws by using volume integral models.In most of
investigated models,the axis of the driving coil is assumed to be coaxial with the axis of the inspected
tube.Some recent works consider the eddy current problem of a conducting cylinder surrounded by an
eccentric circular coil [5,6,7].However these works highlight the computation of the induced field in
the tube’s wall without considering any flaw.The aim of this paper is to present how to solve the for-
ward problem and to describe a computer modelling tool which is able to address any arbitrary three-
dimensional current source placed inside a tube affected by an arbitrary 3D flaw which may varying by
the shape,the size and the place within the tube’s wall.All the computations are considered at eddy cur-
rent frequencies.The model gives the eddy currents distribution within the tube’s wall and the changes
1383-5416/03/$8.00
c
°2003 – IOS Press.All rights reserved
2 Development of a 3D electromagnetic model for eddy Current tubing inspection
in the impedance due to 3Dflaws.Adifferential probe is modelled by considering that the current in one
coil is 180 degrees out of phase with the current in the other coil.
2.The volume integral formulation
A linear,isotropic,non-magnetic and conducting circular tube (permittivity"
0
,permeability ¹
0
,con-
ductivity ¾
0
) of inner radius a,outer radius b is placed along the z-axis of a cylindrical coordinate sys-
tem (r;µ;z).A bobbin coil of inner radius r
1
,outer radius r
2
and length L is placed inside the tube
(Region D
1
).Figure 1 displays the configuration.The probe position might not be coaxial with the tube
and let denote by c the eccentricity.A driving time-harmonic current i(t) of amplitude I
0
and angular
frequency!is applied to the coil.The constitutive properties of the ambient media denoted by D
3
are
quite similar to those of the air.A 3D flaw is considered within the tube’s wall (Region D
2
) by local
changes in the conductivity ¾(r;µ;z).By using Maxwell’s equations,the electrical field vectors are the
solution of the Helmholtz equations:
r£r£E
1
¡k
2
1
E
1
= i!¹
0
J
s
(r);r 2 D
1
(1)
r£r£E
2
¡k
2
2
(r)E
2
= 0;r 2 D
2
(2)
r£r£E
3
¡k
2
3
E
3
= 0;r 2 D
3
(3)
where J
s
(r) denotes the current source density within the source,k
2
i
;i = 1;2;3 denote the propagation
constants in the regions D
i
;i = 1;2;3 such that k
2
1
= k
2
3
=!
2
"
0
¹
0
and k
2
2
= i!¹
0
¾ (r).The electrical
field in region D
2
can be separated in two vectors.The first one represents the incident field due to the
current sources when there is no flaw,the probe not being necessary centered in the tube,the conductivity
denoted by ¾ (r) becomes ¾ (r) = ¾
0
while the second one is the perturbed field E
p
(r) due to the flaw.
The forward problem is therefore treated in two steps.The first one consists in computing the incident
electrical field denoted by E
i
2
(r) in region D
2
due to the currents sources when there is no flaw and
taking into account the eccentricity of the probe.This field is the solution of (2) keeping in mind that
the propagation constant is reduced to k
2
2
(r) =k
2
20
= i!¹
0
¾
0
.Then,the flaw problem can be handled
in the same way as the previous step by considering that the flaw is excited by an equivalent source
J
2
(r) = [¾
0
¡¾ (r)] E
2
(r) which can be seen as a current dipole density.The problem is to determine
this quantity.The electrical perturbation field E
p
(r) =
h
E
2
(r) ¡E
i
2
(r)
i
within the domain Ω satisfies
the Helmholtz equation:
r£r£E
p
¡k
2
20
E
p
= i!¹
0
J
2
(r) = ¡i!¹
0

0
¡¾ (r)] E
2
(r) (4)
The integral solution of this equation involves the electric-electric dyadic’s kernel G
(ee)
ij
(r;r
0
) where
the subscripts i and j denote the region of observation and source respectively.For any unit vector u
among (u
r
;u
µ
;u
z
) defined by the cylindrical coordinate system (O;r;µ;z),the quantity G
(ee)
22
(r;r
0
) ¢ u
corresponds to the electrical field due to a u¡directed unit dipole source located at r
0
.For canonical
geometries,such as plane geometries or cylindrical layered structures,the Green’s functions can be found
in explicit analytical expressions [8].Some numerical difficulties have been overcome by combining
specific approximations of Bessel and Hankel functions [9].The internal electrical field in region D
2
is
obtained by the superposition of the incident field E
i
2
(r) and the perturbed field E
p
2
(r):
Development of a 3D electromagnetic model for eddy Current tubing inspection 3
E
2
(r) = E
i
2
(r) +E
p
2
(r) = E
i
2
(r) +k
2
20
Z
Ω
G
(ee)
22
(r;r
0
)¢ [f (r) E
2
(r)] d
3
r
0
(5)
while Ω being the domain of inhomogeneities and f (r) the contrast function such that f (r) =

0
¡¾ (r)] =¾
0
.By multiplying both sides of equation (5) by ¾
0
f (r),one obtains the integral equation
satisfied by the dipole current density we look for:
J
2
(r) = J
i
2
(r) +k
2
20
f (r)
Z
Ω
G
(ee)
22
(r;r
0
) ¢ J
2
(r) d
3
r
0
(6)
where J
i
2
(r) = ¾
0
f (r) E
i
2
(r) corresponds to the source term.This state equation is the keystone of the
volume integral approach.Finally,the incident electrical field may be expressed by an integral equation:
E
i
2
(r) = k
2
20
Z
coil
G
(ee)
21
(r;r
0
) ¢ J
s
(r)d
3
r
0
(7)
where G
(ee)
21
(r;r
0
) is the dyadic Green’s function corresponding to a source in region D
1
and the field
observed in the region D
2
.The expressions of the current density must be written in the cylindrical
coordinate system(r;µ;z) related to the observation domain.
By using the reciprocity principle [10,11] which involves the excitation vector and the equivalent source
density and including the eccentricity,the changes in the impedance for a single coil due to the flaw is
given by:
±Z = ¡
1
I
2
0
Z
Ω
E
i
2
(r) ¢ J
2
(r) d
3
r (8)
Where I stands for the driving current in the exciting probe.This probe impedance results from the
interaction between the incident field with the flaw.However,in non destructive testing,the experimental
signal is usually balanced by a reference signal measured in an healthy zone with the probe remaining
centered.So,for obtaining the EC signal denoted by ±Z
EC
,we must add the contribution due to the
eccentricity:
I
2
0
±Z
EC
= ¡
Z
Ω
E
i
2
(r) ¢ J
2
(r) d
3
r ¡
Z
coil
£
E
i
2
(r) ¡ E
i
2
(r)
¯
¯
c=0
¤
¢ J
s
(r) d
3
r (9)
where E
i
2
(r)
¯
¯
c=0
stands for the incident field when the eccentricity is null.Most Probes used for inspec-
tions are differential type.Thus,the dipole current density is therefore computed by solving equation (6)
with considering that,by using the principle of superposition,the incident field results from the combi-
naison of each incident field created by each coil,(e.g):E
i
2
(r) =
£
E
i
2
(r)
¯
¯
coil1
¡ E
i
2
(r)
¯
¯
coil2
¤
:Other-
wise,assuming that tilt effects are neglected,the mutually voltage appearing in each coil is theoretically
the same.This last consideration allows to simulate the probe wobble as it often occurs in non destructive
tubing inspection and thus the eddy current signal is given by:
I
2
0
±Z
EC
= ¡
Z
Ω
£
E
i
2
(r)
¯
¯
coil1
¡ E
i
2
(r)
¯
¯
coil2
¤
¢ J
2
(r) d
3
r (10)
4 Development of a 3D electromagnetic model for eddy Current tubing inspection
j
ª
(u
r
;u
µ
;u
z
)
3
6
?
6
L
6
z
-
?
y
-
¾
c
M(r;µ;z)
µ
x
¾
¾
a
1
a
2
-
r
1
-
r
2
6
d
Fig.1.
Schematic of a differential type probe scanning a
conducting tube
−1
−0.5
0
0.5
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
2D configuration at 100 kHz
ℜe( Δ Z ) in V
ℑm( Δ Z ) in V
Outer groove
40% through wall
Inner groove
10% through wall
Fig.2.
Impedance trajectories for a differential type probe
:’–’ experimental data,’.’ 2DModel,’+’ 3DModel
where E
i
2
(r)
¯
¯
coil1
and E
i
2
(r)
¯
¯
coil2
stand for the incident field created respectively by the first coil and
the second coil.The numerical model have been obtained by using the Galerkin variant of the Moment
Method [12].
3.First numerical results and work in progress
The model have been first evaluated for 2D configurations assuming axysymmetric defects and by
comparing data to other simulated data obtained with a 2D computer modelling tool CIVA [2,13] or
then by comparing to experimental data.Let us consider a differential type probe made of two identical
bobbin coils of 2 mm height,of 7:83 mm inner and 8:5 mm outer radii.The coils are separated by
a distance of 2:5 mm center to center.Two circumferential defects are considered:the first one is an
external groove of 1 mmlength,of 40 %through the tube’s wall denoted by GE40 and the second one is
an inner groove of 1 mmlength,10%through the tube’s wall denoted by GI10.All the experimental data
are calibrated on data measurements providing from the GE40 type flaw.the operating frequency is 100
kHz.The tube is characterized by a = 9:83 mm,b = 11:1 mm and ¾
0
= 1 MS/m.The probe scans the
z axis from ¡10 mm to +10 mm with a 0:2 mm sampling rate.Figure 2 displays the impedance plane
trajectories obtained with the experimental data,the 2D code and the 3D model.The CPU time required
for computing the trajectories is fewer than three minutes.Another validation have been carried out for
3Dconfigurations.Figure 3 shows for instance the plane trajectories obtained for a flawof 100 %through
the tube’s wall,of 0:113§0:02 mmlength and of 82
±
extended along the circumferential direction,all the
other parameters remain the same as previously.This figure makes a comparison between simulated data
and experimental data.In this case,the CPU time is fewer than ten minutes.Finally,Figure 4 presents
typical impedance trajectories obtained for three values of the eccentricity c.The considered 3D flaw is
an outer diameter transversal notch with a depth of 20%though wall,a length of 0:1 mmand an angular
extension of 90
o
.The operating frequency is 100 kHz.About forty minutes are required to compute the
trajectories and the time consuming may increases as the value of the eccentricity does.
Development of a 3D electromagnetic model for eddy Current tubing inspection 5
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
−0.1
−0.05
0
0.05
0.1
3D Configuration at 100 kHz
ℜe( Δ Z ) in V
ℑm( Δ Z ) in V
Fig.3.
Impedance trajectories for a differential type probe
:’–’ experimental data,’+’ Model
−5
0
5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Real(ΔZ) in mΩ
Imag(ΔZ) in mΩ
z = +0.0 mm
z = +0.6 mm
z = +1.8 mm
z = +3.0 mm
z = +4.2 mm
z = −4.2 mm
z = −3.0 mm
z = −1.8 mm
z = −0.6 mm
c = − 0.3 mm
c = 0.0 mm
c = + 0.3 mm
Fig.4.
Impedance trajectories for a differential type probe
and for different values of the eccentricity:c =
¡0:6;¡0:3;0;+0:3 mm
4.Conclusion
A 3-D model for eddy current tubing inspection has been developed and a fast numerical code has
been implemented.The code may be used to study the perturbations which can affect the EC signal such
as the variations of the eccentricity.Good agreement have been observed when comparing experimental
data and simulated data for 2D configurations in one hand and a 3D configuration in the other hand.
Further validations will be done for a variety of 3D geometries with laboratory controlled experimental
data.Work is presently in progress to solve the case of an arbitrary shaped and positioned probe placed
inside the tube.
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[1]
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(1968) 2829-2838.
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