## 1 Introduction
### Stray Parameters When Measuring \(d\) by Using the Amplitude-Sensitive Eddy-Current Method  
Stray parameters affecting the value of measurement uncertainty can be divided into the following groups:  
* Geometric: curvature and roughness of a base metal and coating surfaces; the tilt of ECP; contamination of the coating and ECP surfaces resulting in a gap between the ECP and the test object; edge effect. * Technological: the pressure of ECP on the coating; intermediate coatings; the influence of temperature and external electromagnetic fields. * Electromagnetic: the specific electrical conductivity \(\sigma\) of the base metal. Practice shows that the value of \(\sigma\) and its changing along the surface of the base metal and/or from one object to another make the greatest contribution to the measurement uncertainty [3]. To study the influence of the test and stray parameters, one should use the analysis of the ECP parameters on the complex plane. Figure 1: Amplitude-sensitive eddy-current method of measurement of the thickness of coating. 1—probe core with coil W, 2—high frequency magnetic field, 3—non-conductive coating, 4—non-magnetic electrically conductive base metals, 5—induced eddy current, \(I\)—exciting current, \(U\)—signal of eddy-current converter. Figure 2: Structure of eddy-current probe with self-oscillating circuit. The coil impedance \(\overline{Z}\) of a parametric ECP is a complex value determined by the formula:  
\[\overline{Z}=r+jXl \tag{2}\] \[Xl=2\pi fL,\]  
where \(r\) is ohmic resistance and \(Xl\) is inductive resistance. Figure 3 (on the left) shows the dependence of the coil impedance \(\overline{Z}\) of the parametric ECP on \(d\) on the complex plane, Figure 3 (on the right) shows the local hodograph diagrams of the coil impedance of the parametric ECP when changing \(d\) from 0 to \(\infty\) at \(\sigma\) = 5 MS/m, 17 MS/m and 58 MS/m and \(\sigma\) from 0 to 58 MS/m. On placing an ECP on an uncoated base metal (\(d\) = 0) with nominal value \(\sigma\) = 17 MS/m, the impedance on the complex plane corresponds to the point \(d\) = 0, \(\sigma\) = 17 MS/m. With an increase in \(d\), the hodograph diagram of the impedance change describes line 1. When an ECP is placed on an uncoated base metal (\(d\) = 0) with a different specific electrical conductivity; for example, \(\sigma\) = 58 MS/m or 5 MS/m, the impedance on the complex plane corresponds to the point \(d\) = 0, \(\sigma\) = 58 MS/m or \(d\) = 0, \(\sigma\) = 5 MS/m. As \(d\) increases, the hodograph diagram of impedance change is line 2 or 3. In this case, the dependence\(f\) on \(d\) at \(\sigma\) = 5 MS/m, \(\sigma\) = 17 MS/m and at \(\sigma\) = 58 MS/m is described by the curves shown in Figure 4a, and the dependence the deviation of \(d\) on \(\sigma\) on a logarithmic scale is shown in Figure 4b. When measuring \(d\) on base metals with different values of \(\sigma\), a multiplicative change in \(f\) occurs, which can be compensated by adjusting (calibrating) the ECP on an uncoated base metal or a coated base metal with a known thickness \(d\). Such a compensation will be observed in a small range of \(d\) near the calibration point. However, sometimes, we cannot perform a compensation; for example, in the following cases:  
* The thickness is measured on a fully painted product, and removing the coating from a part of the surface (for ECP calibration) is not possible;
* \(\sigma\) changes considerably on the surface of base metal;
* The test object has different temperatures on the surface. In this case, a deviation of \(\sigma\) is observed. Figure 3: Dependence of the coil impedance of the parametric ECP on the thickness of coating and specific electrical conductivity. In these cases, it is necessary to provide the suppression the influence of \(\sigma\) by implementing the adaptive structure of the measuring probe, which has functional redundancy [4; 5]. In indirect measurement methods using the ECP graduation curve, the interpolation error of the graduation curve is an additional source of measurement uncertainty.