A definition of eddy current penetration depth useful for flaw ... - ZARM

Proceedings in Applied Mathematics and Mechanics, 30 April 2008
A definition of eddy current penetration depth useful for flaw detection
and conductivity measurement
Leszek Dziczkowski
1 SILESIAN UNIVERSITY OF TECHNOLOGY, ELECTRONICS FACULTY, Akademicka 16, 44-100 Gliwice, Poland
The proposed definition of the penetration depth of eddy currents, more convenient for tests with defectometers, is based on
the model: measuring coil placed above non-magnetic conductive foil. The designated penetration depth is equivalent to the
penetration depth of a measuring device and is subject of practical observations.
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1 Introduction
The results of tests conducted by devices operating on the bases of eddy currents impact on measuring coil impendance, depend
only on the properties of a definite close-to-surface layer of the tested material. If the penetration depth of eddy currents is
considered, it is automatically associated with the distance designating the space where the intensity of the electromagnetic
field during tests is different from zero. For practical reasons, zero is replaced with the boundary value of the field intensity.
If, at a specific point, the field intensity is lower than the boundary value, this point is beneath the penetration depth of eddy
currents. The designation of the penetration depth on the grounds of the boundary value of the field intensity, assumed a-
priori, or dependent on the accuracy of a measuring device, turns out as erroneous and may lead to apparently inexplicable
phenomena. An example of such phenomena involves the observations made in the course of measuring the conductivity of
non-magnetic plates. The operator, by slowly increasing frequency, may notice that after some successive steps, instead of
decreasing, the penetration depth also increases.
2 The proposed definition of eddy currents penetration depth
The definition of eddy currents penetration depth based on the model of the foil is more useful for defectometers and conductometers.
For simplification, let us assume that the coil has n turns concentrated in a circle with radius r 0 , placed at distance
h from the tested foil with thickness d. The position of the coil in relation to the measured element is shown in Fig.1.
Let us calculate the change in the coil impendance evoked by the presence of the conductive material [1]. For this purpose,
the following generalized parameters are useful:
α = 2h
r 0
β = r 0
√ ωµ0 σ ρ = 2d
r 0
(1)
where: σ - conductivity of the material, ω- current angular frequency in the coil. The change in the coil impendance (∆Z) is
expressed by the following equation: ∆Z = n 2 ωπµ 0 r 0 Q(α, β, ρ) where:
Q(α, β, ρ) =
∫ ∞
0
J 2 1 (y)e −αy β 2 (1 − e −ρ√ y 2 +jβ 2 )
( √ y 2 + jβ 2 − y) 2 e −ρ√ y 2 +jβ 2 − ( √ y 2 + jβ 2 + y) 2 dy (2)
By separating the real and imaginary parts of the equation from the abstract one, it is possible to derive a relation that describes
the change in the coil impendance components evoked by the presence of the conductive foil:
r = R − R 0 = n 2 ωπµ 0 r 0 ϕ(α, β, ρ) (3)
where:
l = L − L 0 = −n 2 πµ 0 r 0 χ(α, β, ρ) (4)
ϕ(α, β, ρ) = RQ(α, β, ρ) (5)
χ(α, β, ρ) = IQ(α, β, ρ) (6)
R 0 and L 0 - are the parameters of the coil distanced from the tested element.
The penetration depth of eddy currents may also be determined as a minimal conductive foil thickness d influencing, in
consideration of the accuracy of measurements, the change in the impendance components to the same degree as any semispace
with identical conductivity. By approaching the coil to the conductive semi-space, its impendance components change
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PAMM header will be provided by the publisher 2
Fig.1. The contact coil above the tested element Fig.2. Graphs of function a(α, β, ρ)
by values: r and l. The penetration depth of eddy currents is regarded as equal to dp designated in the following way: after
approaching an identical coil at the identical distance, not to the semi-space, but, this time, to the foil with thickness d p the
coil impendance components changed by values r and l designated with the accuracy of the error in the measurement by a
given measuring device. Let us denote the error in designating the resistance change as δr, and the coil inductivity as δl. For
given generalized parameters α and β, each of the equations expressed below may be solved in view of unknown ρ.
2 · δr
n 2 ωπµ 0 r 0
=
∣ ∣∣∣ ∣ϕ(α, β, ρ) − lim ϕ(α, β, ρ) ⇒ ϕ r
ρ→∞
2 · δl
n 2 πµ 0 r 0
=
∣ ∣∣∣ ∣χ(α, β, ρ) − lim χ(α, β, ρ) ⇒ ϕ l
ρ→∞
Let us assume that the maximal value from all the calculated values of ρ r and ρ l shall be the generalized penetration depth
of eddy currents: ρ p = MAX(ρ r , ρ l ). The real penetration depth may be derived if the dimensions of the contact coil are
known: d p = (ρ p · r 0 )/2.
3 Some properties of the proposed definition of eddy currents penetration depth
To illustrate the outcome of the proposed definition of eddy currents penetration depth some numerical calculations were
made. In Fig.2 the values of function a(α, β, ρ) = n2 ωπµ 0 r 0
2
· [ϕ(α, β, ρ) − lim ρ→∞ ϕ(α, β, ρ)] were plotted, depending on
parameter ρ for α=0,4 and for some values of β=3, 4, 5 i 6.
To designate the penetration depth of eddy currents, the values of function a(α, β, ρ) should be compared with the error in the
coil resistance measurement. The insensitivity zone of the measuring device was plotted in Fig.2.
For generalized parameter β = 3, generalized penetration depth is ρ 3 . Likewise, if frequency is increased to the value
of generalized parameter β = 4, the penetration depth of eddy currents shall decrease to the value corresponding to ρ 3 .
Another increase in frequency corresponding to the value of generalized parameter β = 5 leads to improved sensitivity of the
measuring method at higher depths and, accordingly, to the penetration depth of eddy currents equal to ρ 5 which is bigger
than ρ 4 . Every successive increase in frequency evokes a significant decrease in the penetration depth of eddy currents to the
values of generalized parameter ρ 6 .
4 Conclusions
Changes in the frequency of the exciting field lead to changes in the penetration depth of eddy currents. The penetration depth
may be calculated by means of numerical methods on the grounds of the model: contact coil placed above semi-space. It
turns out that the presence of a flaw in the tested material, even if located beneath the designated boundary, is detected by the
measuring device. Such flaw apparently increases the penetration depth of eddy currents. Thus, the measuring device has a
penetration zone bigger than the calculated eddy currents penetration depth. The proposed definition: measuring coil placed
above conductive non-magnetic foil is more convenient for defectometers, as it is equivalent to the penetration depth of the
measuring device. On the grounds of the described model, every advanced piece of equipment may be used for estimating and
displaying the actual penetration depth of eddy currents, depending on the coil properties and frequency.
References
[1] L. Dziczkowski, M. Dziczkowska, A useful mathematical model for analysis of non-magnetic thin foil on the grounds of the eddy
current method. Mainostroenie i Technosfera XXI weka. Sprawnik Trudow XIV Medunarodnoj Naucno-techniceskoj Konferencji,
Donieck-2007, T.5, p.26-31.
[2] Dziczkowski L., The analysis of determining the conductance and thickness of thin non-magnetic foil by the eddy current method,
Mainostroenie i Technosfera XXI weka. Sprawnik Trudow XIV Medunarodnoj Naucno-techniceskoj Konferencji, Donieck-2007, T.5,
p.22-26.
[3] Dziczkowski L., Errors in the simultaneous determination of conductivity and foil thickness by the eddy curent method based on a
single measurement, Avtomatizacja: Problemy, Idei, Reenija, Materialy Medunarodnoj Naucno-techniceskoj Konferencji, Sevastopol-
2007, p.137-140.
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