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.
structuroscopes - to determine the structural state of the TO as a result of fixing the depth
profiling of electrical conductivity and/or magnetic permeability [
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 [
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-tonoise ratio (SNR). Recently, the method of robust parameter design [
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
[
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.
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.
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.
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.
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:
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: (5) (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 GaussLaguerre 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.
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.
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).
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.
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). 14 15 6 7 8 9 0 1 2 32 42 52 62
1 1 1 1 2 2 2 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00 ,00
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.
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.
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].
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. [14] T. Orosz, A. Rassõlkin, A. Kallaste, P. Arsénio, D. Pánek, J. Kaska, P. Karban, Robust Design Optimization and Emerging Technologies for Electrical Machines: Challenges and Open Problems, Applied Sciences, 10(19) 2020, 6653. doi:10.3390/app10196653. [15] V. S. Sobolev, Y. M. Shkarlet, Surface and Screen Sensors [in Russian]. 1967. [16] F. C. Wu, Simultaneous optimization of robust design with quantitative and ordinal data.
International journal of industrial engineering: Theory, applications and practice, 15(2) (2008), 231-8. doi:10.23055/ijietap.2008.15.2.124. [17] S. Hernández, J. Díaz, An application of Taguchi’s method to robust design of aircraft structures. High Performance Structures and Materials VI, 124 2012, 3-12. doi:10.2495/HPSM120011. [18] V. Y. Halchenko, R. V. Trembovetska, V. V. Tychkov, Development of excitation structure RBF-metamodels of moving concentric eddy current probe, Electrical Engineering & Electromechanics, (2) 2019. 28–38. doi:10.20998/2074-272X.2019.2.05 [19] V. Y. Halchenko, R. Trembovetska, V. Tychkov, Surrogate synthesis of frame eddy current probes with uniform sensitivity in the testing zone, Metrology and measurement systems, 28(3) 2021, 551-564. doi:10.24425/mms.2021.137128.