PAMM · Proc. Appl. Math. Mech. 9, 187 – 188 (2009) / DOI 10.1002/pamm.200910068
Selection of the eddy currents frequency for conductivity measurements
in two-layer structures
Leszek Dziczkowski ∗1
1
Silesian University of Technology, Faculty of Electronics, Akademicka 16, 44-100 Gliwice, Poland
Conductance measurements of conductive parts can serve as a background for conclusion about structure of material and
possible defects (e.g. fractures, fissures, delamination, flaking, etc.) that may occur in examined parts. When the need arises
to test surfaces of parts that are made of a conductive but non-ferromagnetic material, the method of eddy currents offers
unsurpassed advantages. The problem refers to application of the eddy current method to examination of conductive material
coated with conductive films. The major outcome consists in development of a very simple and useful mathematical model
that can be used to determine conductivity of the deep layer for the needs of non-destructive tests.
c 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
The appearing variations in conductance hat can be detected with use of the eddy-current method brings information about
surface defects or about alterations of the material structure. In order to efficiently examine damage of materials and detect
various fractures as well as properly construe indications of instruments it is necessary to struggle for such conditions that
guarantee the highest possible accuracy of conductance measurements. Results of experiments with use of instruments with
their operation principle based on the effect of eddy currents onto impedance of the measuring coil depend on parameters of
only a small, surface-adjacent layer of the examined material. When the research is focused on investigation of a conductive
material coated with a conductive film, the outermost layer significantly prevents from penetration of the magnetic field deeply
into the structure. In such a case any conclusions on electric parameters or continuity of deeper layers exhibit high degree of
uncertainty. Hence, it is necessary to provide such measurement conditions when vulnerability of the measuring instrument to
variations of parameters attributable to the deeper layer is the highest with minimum influence of deviations in parameters of
the outer film. Outcomes and conclusions from the study can be useful for both designers of measuring instrument that take
advantage of the eddy current effect and users that deal with measurements of flaw detection.
2
Mathematical model
The examined structure is made of a non-ferromagnetic metal with conductivity σp . Surface of the structure is coated with a
conductive film with its thickness of d and conductivity of σi . The real contact coil that is usually used for measurements is
substituted with a model coil with all n of its turns encapsulated by a circle with the radius of r0 disposed within the distance of
h from the examined surface. Mutual location of the coil and the examined structure is presented in Fig. 1. When alternating
current with angular frequency of ω flows through that coil, the magnetic field produced by that coil induces eddy currents in
the examined material. These eddy currents produce own electromagnetic field that counteracts the external field and reduces
its value. Eventually, [1] impedance of the coil shall change by ∆Z.
∞
2
where : Q(α, β, ρ, s) = jβ
c(α, β, ρ, s)J12 (βy)e−αβy dy
(1)
∆Z = n ωπµ0 r0 Q(α, β, ρ, s)
0
The function c(α, β, ρ, s) is defined by the formula:
√ 2
( y 2 + js + y 2 + j)( y 2 + j − y) + ( y 2 + js − y 2 + j)( y 2 + j + y)e−ρβ y +j
√ 2
c(α, β, ρ, s) = (2)
( y 2 + js − y 2 + j)(y − y 2 + j)e−ρβ y +j − ( y 2 + js + y 2 + j)(y + y 2 + j)
In practice, it is convenient to use generalized parameters:
2h
2d
σp
√
β = r0 ωµ0 σi
ρ=
s=
r0
r0
σi
Variations of components that make up the coil impedance can be calculated from the following relationships:
α=
r = ∆R = R − R0 = n2 ωπµ0 r0 · ϕ(α, β, ρ, s)
∗
where : ϕ(α, β, ρ, s) = Q(α, β, ρ, s)
(3)
(4)
Corresponding author: e-mail: ldziczk@wp.pl
c 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
188
Short Communications 3: Damage and Fracture
Fig.1. The contact coil approached to a conductive, twolayer structure
l = ∆L = L0 − L = n2 πµ0 r0 · χ(α, β, ρ, s)
Fig.2. Examples of diagrams for impact coefficients that
define relationship between conductivity of the deep layer
and variations of the measuring coil resistance
where : χ(α, β, ρ, s) = − Q(α, β, ρ, s)
(5)
R0 and L0 stand for resistance and inductance of the coil positioned within a distance from the examined structure. The
formulas (4) and (5) make it possible to calculate impact coefficients that exhibit how variations of conductivities σp and σi
affect variations in resistance and inductance of the measuring coil.
∆rσp =
∆R
n2 πβ 2 ∂ϕ
=
∆σp
r0 σi2 ∂s
(a)
∆rσi =
∆R
n2 πβ 2
=
∆σi
r0 σi2
∆lσp =
∆L
n2 πr0 µ0 ∂χ
=
∆σp
σi
∂s
(a)
∆lσi =
∆L
n2 πr0 µ0
=
∆σi
σi
β ∂ϕ σp ∂ϕ
·
−
·
2 ∂β
σi ∂s
β ∂χ σp ∂χ
·
−
·
2 ∂β
σi ∂s
(b)
(6)
(b)
(7)
The formula (6a) describes how conductivity of the deep layer is reflected in resistance variations of the measuring coil,
whereas the formula (7a) refers to the influence of that conductivity in the deep layer onto inductance variations of that coil.
Respective formulas (6b) and (7b) explain effect of conductivity of the surface film onto variations in resistance and inductance
of the coil. Eddy currents induced in the surface film substantially hold back penetration of electromagnetic fields into deeper
layers of the examined structures. Therefore measurements of conductivity in that lower ply are pretty difficult, but they are
possible anyway. Not only a dedicated measuring workbench must be used, but also the alternating frequency of the field
that induces eddy currents must carefully selected. Analysis of functions defined by formulas (6a), (6b), (7a) and (7b) make
it possible to notice that careless selection of frequency for the exciting field may lead to situation where measurements are
burdened by a substantial error. However, it is enough to amend the frequency a little to achieve satisfying sensitivity. Fig. 2
presents diagrams for the function (6a) with the variable parameter of β and for several values of ρ.
3
Conclusions
The determined impact coefficients, defined by the relationships (6a), (6b), (7a) and (7b) perfectly suit to determine the
optimum frequency of eddy currents. To do this one can find extreme value of the appropriate function for any specific case.
Equations (6a) and (7a) enable also to find out limits that restrict applicability of the eddy current method for examination
of deep layers. For instance, let us denote the relative error for measurements of the coil resistance as δR and δσp shall
stand for the permissible relative error for measurements of the conductivity with the exact value of σp . The permissible
error is the maximum deviation of the measurement from the exact value that is still acceptable. The value of βbo that is
graphically determined on the drawing corresponds to such frequency of eddy currents, for which the error for measurement
of conductivity in the deep layer is the lowest and equal to the maximum permissible value, whereas the deep layer is covered
by a film with its thickness of ρ = 0.125. Thus, all measurements, when the thickness of the outer film if higher than
ρ = 0.125, shall only increase the error value above the permissible threshold, which makes the measurement pointless.
References
[1] L. Dziczkowski, M. Dziczkowska, Hindrances associated with examination of two-layer structures with use of the eddy current
method, Archves of Materials Science and Engineering, Volume 35, Issue1, January 2009, pp. 39-46.
c 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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