Computer robust parameter design of surface
eddy current probes. Developing software
design of numerical experiments creating ⋆
Volodymyr Halchenko1,†, Ruslana Trembovetska1,† and Volodymyr Tychkov1*,†
1
Cherkasy State Technological University, 460 Shevchenko Blvd., Cherkasy, 18006, Ukraine
Abstract
In this study, the initial stage of computer robust parameter design of a surface eddy-current probe on
the example of a thickness gauge was performed using the integration of the Taguchi method with
numerical modeling. It involves the selection of controllable design and operating parameters of the
probe and uncontrollable noise parameters. The software for calculating the output signal of the
thickness gauge was created and verified. In order to establish the boundary values of the factors,
numerical modeling was performed, which allowed to obtain graphical dependencies of the change in
the output signal of the probe on the variation of the selected factors. Based on the orthogonal arrays,
taking into account the selected factors, a design of numerical experiments was created that allows
creating robust parameter design using the developed software. Without eliminating the real causes
of interference in the formation of the probe output signal, it ensures the selection of a rational variant
of the set of its design and operating parameters, which implements the minimum variability of the
probe response to noise factors at the initial design stage.
Keywords
computer robust parameter design, Taguchi's complex method, orthogonal array, numerical
experiments design, eddy current probe 1
1. Introduction
It is a well-known fact that the use of eddy current probes (ECPs), in particular, surface ones, in
non-destructive testing is based on the multiparameter nature of the information to be selected.
This provides many opportunities for measuring a significant number of information
parameters.
However, determining one of the specific controllable parameters leads to problems
associated with overcoming the influence of uncontrollable ones, which also form the output
signal of the probes and are essentially noise. Therefore, ECPs can be used for various purposes
as part of flaw detection material integrity violations in the testing objects (TO) [1]; in
⋆
ITTAP’2024: 4th International Workshop on Information Technologies: Theoretical and Applied Problems,
November 20–22, 2024, Ternopil, Ukraine, Opole, Poland
1∗
Corresponding author.
†
These authors contributed equally.
v.halchenko@chdtu.edu.ua (V. Ya. Halchenko); r.trembovetska@chdtu.edu.ua (R. V. Trembovetska);
v.tychkov@chdtu.edu.ua (V. V. Tychkov)
0000-0003-0304-372X (V. Ya. Halchenko); 0000-0002-2308-6690 (R. V. Trembovetska); 0000-0001-9997-307X (V.
V. Tychkov)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
structuroscopes - to determine the structural state of the TO as a result of fixing the depth
profiling of electrical conductivity and/or magnetic permeability [2, 3], in controlling
mechanical stresses [4], chemical composition, quality of technological processing of parts by
chemical and physical methods, and structural anomalies; in thickness gauges - to control the
geometric dimensions of the TO [5] and the coatings' thickness [6, 7].
Depending on the intended purpose of the measurement, in each of the above cases of ECP
application, the above-mentioned factors perform different functions from assisting to
hindering, they were constantly exchanging roles. Each of these examples involves the use of
special techniques and methods for suppressing noise signals, which are often based on the
analysis of the dependence of the ECP output signal on a number of factors, with an attempt to
separate their influence [1, 2, 3], but this approach is quite difficult to implement if it is necessary
to suppress more than one of them. Other methods of the same purpose are known, which are
used either separately or in combination, in particular, stabilization of testing conditions,
application of spectral analysis, etc. However, these techniques for selecting useful information
from the ECP signal are either not fully perfect or rather complicated in practical
implementation, which does not add to their effectiveness. This is especially true in
multiparameter measurement cases.
Thus, the noted limitations of selecting useful information from the ECP necessitate the
search for other approaches to solving this problem that would provide an increased signal-to-
noise ratio (SNR). Recently, the method of robust parameter design [8, 9] has become quite
popular among researchers, which has attained wide application in various fields of
instrumentation and measurement technology. The point is that this effect can be achieved at
the initial stage of ECP computer design, and not during the selection and processing of the
probe signal. The ECP measurement process is characterized by uncertainties such as changes in
the lift-off, local changes in the electrophysical properties of the material, local variations in the
geometry of the TO, possible imperfections of its surface, such as roughness, curvature, etc. In
such conditions, the robust parameter design of the ECP provides maximum sensitivity to the
controllable parameter, while for other influential, but interfering with measurements, the
sensitivity is minimal. Therefore, robust parameter design can be used to effectively select
rational design and operating parameters of the ECP, which provide the probes with resistance
to noise caused by uncontrollable variations, i.e., robustification is aimed at numerical finding
parameters of controllable factors that minimize the deviation of the response from
uncontrollable ones due to the use of nonlinearity of their effect on the signal by computing
facilities. Robust parameter design is usually based on experimental data, but it can also be
obtained by computer modeling as a result of numerical experiments, and therefore is an
engineering methodology. Its result is achieved by reducing the effects of variations without
actually eliminating their causes. A numerical indicator of successful robustification in
accordance with the proposed design computer concept can be considered a criterion that
requires maximization (quality loss function “larger-the better”) by the choice of factor levels
[10] and ensures the largest value of the ECP signal and minimization of its variability, i.e.,
variance, and corresponds to the expression:
(1)
,
where n is the sample size, E is the ECP signal.
Despite the widespread use of robust parameter design based on the Taguchi method in
various fields of science and technology, as evidenced by quite old publications, in particular
[11], the authors have not found any studies on its application in the design of ECPs. At the same
time, the relevance of the results of its use has not been lost in the present, as evidenced, for
example, by articles [12, 13, 14]. Significant practical results of its application with the
involvement of insignificant computational resources and the absence of interventions to level
the effect of noise factors are undeniable.
Thus, the purpose of the article is to create a methodology for computer robust parameter
design of surface eddy current probes and related software based on the integration of the
Taguchi method and numerical modeling, which allows, at the initial stage of choosing its
design, to achieve selective adjustment of the probes` sensitivity to measure the useful signal
while reducing it to uncontrollable interfering factors.
2. Research methodology
In order to implement computer robust parameter design of the ECP, a number of steps are
required according to the Taguchi method. A general scheme illustrating the entire design
process is shown in Fig. 1.
Figure 1: General scheme of the robust parameter design process of the ECP.
The Taguchi method uses special orthogonal arrays to design of experiment and analyze the
resulting data using the SNR. For further implementation of this algorithm, a thorough
understanding of the measuring process, for example of the TO’s thickness, which, from the
general view shown in Fig. 2 a, should be interpreted in the appropriate terms of eddy current
determination of the signal parameter (Fig. 2 b).
understand the relationship observed between the input measured parameter and the target
characteristic of the SNR ECP, taking into account the influence of controllable, i.e., design and
operating factors, and uncontrollable, i.e., noise sources. The input signal is subject to changes
with the corresponding observation of the output response, which allows us to investigate the
value of the controllable factors, the combination of which ensures the smallest possible
variability of the output response.
Figure 2: Measurement process: a - in the general case; b - by eddy current probe.
Computer modeling is used in these studies to create numerical design of experiments.
The electrodynamic mathematical model for a coil with an alternating sinusoidal current
of angular frequency ω, which describes the process of eddy current measurement by a
surface probe over an TO in the form of a conductive plate of finite thickness (Fig. 3), was
obtained as a result of analytical solution of the partial differential boundary value problem in
the cylindrical coordinate system [15]:
(2)
,
where - is the azimuthal component of the magnetic vector potential, Wb/m; ρ, z –
coordinates, m; µ0 = 4·π·10-7 is the magnetic constant in vacuum, H/m; µ - is the relative
magnetic permeability of the medium; jex - density of currents of external sources, A/m2; σ - is
the electrical conductivity of the medium, S/m.
Figure 3: Geometric model of the eddy current measurement process.
Equation (1) was solved on the following boundary conditions:
(3)
,
where p is the number of the boundary of media distribution, р = 1, 2.
The solution was found through the following assumptions: the probe field is considered
quasi-stationary; wave processes in the air are neglected; bias currents in the conductive
medium are also neglected; the diameter of the coil cross-section is considered very small. Under
these conditions, the magnetic vector potential in the area of the ECP pick-up coil can be
determined by the formula:
(4)
,
where
J1() is a first-order Bessel function of the first kind.
If we assume that the real ECP excitation coil has finite geometric dimensions, then to take
into account its cross-section (R2-R1)(h2-h1), the formula for calculating the magnetic vector
potential will be found by integration according to the expression:
(5)
Thus, the output signal of the surface ECP in the form of an EMF induced in the pick-up coil
can be calculated according to the formula:
(6)
,
where wmes is the number of turns of the pick-up coil; P is the observation point with
coordinates (ρ, z) belonging to the contour Lc of the pick-up coil.
To calculate the non-proprietary integral of the first kind (4), it makes sense to use the Gauss-
Laguerre quadrature formula. The creation of the design of experiment involves the calculation
of the ECP EMF for numerical modeling of the measurement process, for which the
corresponding software was developed in the PTC MathCAD Prime environment. Its
verification for the case of representing the probe excitation system by a coil (4) was carried out
in the software environment for solving and simulating various engineering applications
COMSOL Multiphysics using the finite element method. The grid model for this numerical
experiment is shown in Fig. 4, and Fig. 5 shows the results of calculations of the vector magnetic
potential performed at a set of observation points.
Figure 4: Grid model of the surface ECP
The test numerical simulation was performed with the following input data: (ρ, z) = (10·10-3,
1·10-3) m; f = 2 kHz; d = 5·10-3 m; R = 20·10-3 m; h = 2·10-3 m; I = 1 A; = 3.77·107 S/m, µ = 1.
Comparison of the results of calculating the values of the vector potential obtained in the
COMSOL Multiphysics environment (Fig. 5) and in the PTC MathCAD Prime environment
according to formula (4) gives a coincidence of the vector potential values with an accuracy of
0.039 %, which indicates the adequacy of the created software for modeling of the measurement
process.
Consequently, it becomes possible to set the lower and upper limits of variation of all
influencing factors by modeling, that is, to fulfill the task of block 1 of the general scheme of the
robust parameter design process of the ECP. This, in turn, allows choosing the type of
orthogonal array and complete the creation of numerical design of experiments for further
computer robust parameter design.
Figure 5: Results of test calculations of the vector magnetic potential.
3. Numerical experiments
For the purpose of further research, we will limit ourselves to considering the example of an
eddy current thickness gauge, while similar actions are assumed for other measurements. The
analysis of the physical process of thickness measuring of the TO of the ECP allows to identify
the following influencing factors on the output signal of the probe, including controllable (C),
noise (N), or uncontrollable and signal (S) (Table 1).
Table 1
Influencing factors on the output signal of the ECP during thickness measurement of the testing
object
Factor Type of factor Type of parameter
Internal radius of the excitation coil R1 С
External radius of the excitation coil R2 С
Radius of the pick-up coil ρ С Structural
Height of the pick-up coil z С
Distance to the top edge of the excitation coil h2 C
Excitation frequency f С
Mode
Excitation current I C
Magnetic permeability µ N
Electrical conductivity σ N Noise
Lift-off h1 N
TO thickness d S Signal
Subsequently, to establish the sensitivity of the probe to measuring the useful signal, a series
of numerical experiments were performed to determine the dependence of the ECP output
signal on the influencing factors using formula (6). In this case, in each individual experiment,
the factor under analysis varied within certain specified limits, while all other factors remained
unchanged, i.e. fixed. The initial data for this analysis are as follows: R1=20·10-3 m, R2=21·10-3
m, h1=2·10-3 m, h2=3·10-3 m, z=1·10-3 m, r=13·10-3 m, d=3·10-3 m, f=1.5 kHz, I=1 A, σ=6.99·106
S/m, µ=20.
Fig. 6 shows the graphs of changes in the ECP signals when varying such design parameters
as the internal R1 and external R2 radii of the excitation coil. In this case, taking into account the
initial input data, the variation of the inner radius is set in the range
, and the outer radius -
respectively.
-1,900 -1,920 R2-0,4%R2
R1-8%R1
R2=21 mm
-1,905 -1,925
R1-6%R1 R2+2%R2
-1,910 -1,930
R1-4%R1 R2+4%R2
arg(E), rad
arg(E), rad
-1,915
-1,935 R2+6%R2
R1-2%R1
-1,920
R1=20 mm -1,940 R2+8%R2
-1,925
R1+2%R1 -1,945 R2+10%R2
-1,930
R1+4%R1 -1,950 R2+12%R2
-1,935
-1,940 -1,955
0,0190 0,0200 0,0210 0,0220 0,0175 0,0185 0,0195 0,0205
0,0195 0,0205 0,0215 0,0225 0,0180 0,0190 0,0200 0,0210
mod(E), V mod(E), V
Figure 6: Output signal of the eddy current thickness gauge when changing the design
parameters: a - inner radius of the excitation coil R1; b - outer radius of the excitation coil R2.
Figure 7 shows the dependence of the probe signal on the change in the height of the pick-up
coil z and its radius ρ.
-1,920 -1,78
z+10%z
-1,921 -1,80
z+8%z
-1,922 z+6%z -1,82
-1,923 z+4%z
-1,84
arg(E), rad
z+2%z
arg(E) , rad
-1,924
-1,86
-1,925 z=1 mm
z-2%z
z-4%z -1,88
-1,926
z-6%z -1,90
-1,927
z-8%z mm
-1,928 z-10%z -1,92
-1,929 -1,94
0,02030
0,02032
0,02034
0,02036
0,02038
0,02040
0,02042
0,02044
-1,96
0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08
mod(E), V mod(E), V
Figure 7: Dependence of the probe output signal on the change in the design parameters of the
pick-up coil: a - height z; b - radius ρ.
The next design parameter, namely the lift-off h1, was set within
, while simultaneously ensuring the condition of constancy of
the coil cross-sectional area (h2-h1=const). The variation of the ECP signal from the variation of
the lift-off h1 and the distance to the upper edge of the excitation coil h2 and the thickness of the
TO d is shown in Fig. 8.
-1,921 -1,9220 h2+8%h2
h1+8%h1
h1+6%h1 -1,9225
-1,922 h2+6%h2
-1,9230
h1+4%h1 h2+4%h2
-1,923 -1,9235
h1+2%h1
-1,9240 h2+2%h2
arg(E), rad
-1,924
arg(E), rad
h1=2 mm h2=3 mm
-1,9245
-1,925 h1-2%h1 -1,9250 h2-2%h2
-1,926 h1-4%h1 -1,9255
h2-4%h2
-1,927 h1-6%h1 -1,9260
h2-6%h2
-1,9265
-1,928 h2-8%h2
h1-8%h1 -1,9270
-1,929 -1,9275
0,02032
0,02033
0,02034
0,02035
0,02036
0,02037
0,02038
0,02039
0,02040
0,02041
0,02042
0,02033
0,02034
0,02035
0,02036
0,02037
0,02038
0,02039
0,02040
0,02041
0,02042
mod(E), V mod(E), V
-1,923
d+15%d d+25%d
d-2%d
-1,924 d=3 mm
d-6%d
-1,925 d+20%d
d+2%d d+10%d
-1,926
arg(E), rad
d-10%d d+6%d
-1,927
-1,928 d-15%d
-1,929 d-20%d
-1,930
-1,931 d-25%d
-1,932 0,02035
0,02036
0,02037
0,02038
0,02039
0,02040
0,02041
0,02042
mod(E), V
Figure 8: Dependence of the output signal of the eddy current thickness gauge on changes in
the parameters: a - lift-off h1; b - distance to the upper edge of the excitation coil h2; c - thickness
of the TO d.
The following graph (Fig. 9) demonstrates the dependence of the output signal of the probe
on changes in uncontrollable factors, in particular, the magnetic permeability and electrical
conductivity of the TO.
-1,897 -1,897
-1,898 -1,898
-1,899 -1,899
-1,900
-1,900
arg(E), rad
arg(E), rad
-1,901
-1,901
-1,902
-1,902
-1,903
-1,903 -1,904
-1,904 -1,905
-1,905 -1,906
-1,906 -1,907
0,02120
0,02125
0,02130
0,02135
0,02140
0,02145
0,02150
0,02155
0,02160
0,02115
0,02120
0,02125
0,02130
0,02135
0,02140
0,02145
0,02150
0,02155
0,02160
mod(E), V mod(E), V
Figure 9: Output signal of the eddy current thickness gauge when changing uncontrolled
factors: a - electrical conductivity σ; b - magnetic permeability µ.
In conclusion, we studied the change in the ECP signal to varying the operating parameters
of the excitation coil and illustrated the results obtained (Fig. 10).
-1,895 -1,9220
f-20%f
-1,900 -1,9225 I+25%I
f-25%f f-15%f I-25%I
-1,905 -1,9230 I-20%I I+20%I
f-10%f
-1,910 -1,9235
I-15%I I+15%I
-1,915 f-6%f -1,9240
I=1 A
arg(E), rad
arg(E), rad
f-8%f
f-2%f -1,9245
-1,920
f-4%f f+2%f -1,9250
-1,925 I-10%I I+10%I
f=1,5 kHz f+6%f -1,9255 I+8%I
-1,930 I-8%I
f+4%f f+10%f -1,9260 I+6%I
-1,935 I-6%I
f+8%f -1,9265 I-4%I I+4%I
f+20%f
-1,940 -1,9270
f+15%f I-2%I I+2%I
-1,945 -1,9275
f+25%f
-1,950 -1,9280
0,015 0,017 0,019 0,021 0,023 0,025
0,014
0,015
0,016
0,017
0,018
0,019
0,020
0,021
0,022
0,023
0,024
0,025
0,026
0,016 0,018 0,020 0,022 0,024
mod(E), V mod(E), V
Figure 10: Output signal of the eddy current thickness gauge when varying the operating
parameters of the excitation coil: a - frequency f; b - current I.
Based on the obtained graphs (Fig. 6 - Fig. 10), it is possible to numerically determine the
sensitivity of the ECP and determine, respectively, the lower and upper limits of change of each
influencing factor.
Thus, the numerical values of the limits of change of the factors, finally determined by the
graphs, are shown in Table 2.
Table 2
Influencing factors on the output signal of the ECP when measuring the thickness of the TO
Limits of change of influencing factors Lower Upper Factor
bound bound symbol
Inner radius of the excitation coil R1, m 0.0184 0.0208 A
Outer radius of the excitation coil R2, m 0.020916 0.02352 B
Radius of the pick-up coil ρ, m 0.01118 0.0195 C
Distance to the top edge of the excitation coil h2, m 2.76·10-3 3.24·10-3 D
Height of the pick-up coil z, m 9·10-4 1.1·10-3 E
Excitation frequency f, kHz 1.125·103 1.875·103 F
Excitation current I, A 0.75 1.25 G
Magnetic permeability µ 18.4 21.6 H
Electrical conductivity σ, S/m 6.431·106 7.549·106 J
Lift-off h1, m 1.84·10-3 2.16·10-3 K
The orthogonal array L18(21,37) was chosen for the controllable factors, and the array L9(34)
for the uncontrollable ones with three levels of gradation for both types (Table 3) [16, 17].
Subsequently, modernized orthogonal arrays were used, in which one factor was removed, in
particular, a factor with two gradations was removed in L18(21,37), and one extra factor was
removed in L9(34). The values of the factors in the selected orthogonal arrays are converted into
units of real physical quantities (Table 4), corresponding to low, medium, and high levels. The
total amount of computational experiments to be performed according to this design is obtained
by combining the arrays L18(21,37) and L9(34), i.e., 18·9=162. Thus, for each experiment, the
EMF of the probe is determined at the specified settings for each level of all factors.
Table 3
Orthogonal arrays L18(21,37) for controllable and L9(34) for uncontrollable factors
№ Factor controllable Factor uncontrollable
experiment A B C D E F G H J K
1 1 1 1 1 1 1 1 1 1 1
2 1 2 2 2 2 2 2 1 2 2
3 1 3 3 3 3 3 3 1 3 3
4 2 1 1 2 2 3 3 2 1 1
5 2 2 2 3 3 1 1 2 2 2
6 2 3 3 1 1 2 2 2 3 3
7 3 1 2 1 3 2 3 3 1 2
8 3 2 3 2 1 3 1 3 2 3
9 3 3 1 3 2 1 2 3 3 1
10 1 1 3 3 2 2 1
11 1 2 1 1 3 3 2
12 1 3 2 2 1 1 3
13 2 1 2 3 1 3 2
14 2 2 3 1 2 1 3
15 2 3 1 2 3 2 1
16 3 1 3 2 3 1 2
17 3 2 1 3 1 2 3
18 3 3 2 1 2 3 1
Table 4
Design of experiment according to Taguchi method
№ Factor controllable
A B C D E F G
1 0.0184 0.020916 0.01118 0.00276 0.0009 1125 0.75
2 0.0184 0.022218 0.01534 0.003 0.001 1500 1
3 0.0184 0.02352 0.0195 0.00324 0.0011 1875 1.25
4 0.0196 0.020916 0.01118 0.003 0.001 1875 1.25
5 0.0196 0.022218 0.01534 0.00324 0.0011 1125 0.75
6 0.0196 0.02352 0.0195 0.00276 0.0009 1500 1
7 0.0208 0.020916 0.01534 0.00276 0.0011 1500 1.25
8 0.0208 0.022218 0.0195 0.003 0.0009 1875 0.75
9 0.0208 0.02352 0.01118 0.00324 0.001 1125 1
10 0.0184 0.020916 0.0195 0.00324 0.001 1500 0.75
11 0.0184 0.022218 0.01118 0.00276 0.0011 1875 1
12 0.0184 0.02352 0.01534 0.003 0.0009 1125 1.25
13 0.0196 0.020916 0.01534 0.00324 0.0009 1875 1
14 0.0196 0.022218 0.0195 0.00276 0.001 1125 1.25
15 0.0196 0.02352 0.01118 0.003 0.0011 1500 0.75
16 0.0208 0.020916 0.0195 0.003 0.0011 1125 1
17 0.0208 0.022218 0.01118 0.00324 0.0009 1500 1.25
18 0.0208 0.02352 0.01534 0.00276 0.001 1875 0.75
Table 5
Design of experiment according to Taguchi method
№ Factor uncontrollable
H J K
1 18.4 6431000 0.00184
2 18.4 6990000 0.002
3 18.4 7549000 0.00216
4 20 6431000 0.002
5 20 6990000 0.00216
6 20 7549000 0.00184
7 21.6 6431000 0.00216
8 21.6 6990000 0.00184
9 21.6 7549000 0.002
Thus, the obtained numerical design of experiments allows for computer robust parameter
design of the eddy current thickness gauge by selecting rational combinations of the probe's
design and operating parameters. The design quality largely determines the effectiveness of the
ECP design, which has already been proven by the authors' personal experience in applying
other similar data-driven design methods that use designs of experiment [18, 19].
4. Conclusion
Thus, the study carried out, using the example of a thickness gauge, the initial stage of computer
robust parameter design of a surface eddy current probe based on an integrated approach of
combining numerical modeling with the Taguchi method. The physical process of thickness
measurement by the probe is analyzed and, as a result, their controllable and uncontrollable
factors are identified from all possible influencing factors.
Graphs of the output signals of the ECP before changing the parameters of the influencing
factors were obtained and their lower and upper limits were determined, respectively. Taking
into account the number of relevant factors, two types of orthogonal arrays were selected,
namely L18(21,37) and L9(34) with three levels of their gradation.
The software for the implementation of computer robust parameter design of surface ECP
was created, and its verification in the COMSOL Multiphysics environment for calculating the
ECP EMF was carried out, which allows to fully complete the creation of a numerical design of
experiments.
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