# Eddy Current Testing of Conductive Coatings Using a Pot-Core Sensor

## Abstract

**:**

## 1. Introduction

## 2. Analytical Model

_{1}, a bond coating with a thickness of l

_{2}− l

_{1}, and a substrate with a thickness of l

_{3}− l

_{2}. The magnetic permeability of conductive coatings was determined as μ

_{6}, μ

_{7}, and the electrical conductivity as σ

_{6}, σ

_{7}. The problem was analysed in a cylindrical coordinate system, and the solution domain was divided into 9 regions and limited to the value of parameter b (Figure 2). Bounding the solution domain, i.e., limiting the range of a coordinate, results in discrete eigenvalues for that coordinate direction [47]. The discrete eigenvalues

**q**of regions with a homogeneous structure (1, 5–8) are the positive real roots of the Bessel function of the first kind J

_{1}(x) and are calculated from equation J

_{1}(

**q**b) = 0. Region 3 consists of 3 subregions (0 ≤ r ≤ a

_{1}, a

_{1}≤ r ≤ c

_{2}, and c

_{2}≤ r ≤ b). The discrete eigenvalues

**m**of region 3 are the positive real roots of the equation ${L}_{1}^{\prime}(m\hspace{0.17em}b)=0$, where:

_{f}. Regions 3 and 4 consist of 5 subregions (0 ≤ r ≤ a

_{1}, a

_{1}≤ r ≤ a

_{2}, a

_{2}≤ r ≤ c

_{1}, c

_{1}≤ r ≤ c

_{2}, and c

_{2}≤ r ≤ b). The eigenvalues

**p**of these regions were determined—using the Bessel function Y

_{n}(x)—from the equation:

_{0}were placed at a distance h

_{0}from the three-layer conductive structure. At first, the magnetic vector potential for the filamentary coil (r

_{2}− r

_{1}→ 0, h

_{2}− h

_{1}→ 0) was written in the matrix notation:

**s**

_{i}= (

**q**

^{2}+ j ω μ

_{i}μ

_{0}σ

_{i})

^{1/2}, and

**B**

_{i},

**C**

_{i}are the unknown coefficients.

**B**

_{i},

**C**

_{i}:

**B**

_{i8}=

**B**

_{i}/

**B**

_{8},

**C**

_{i8}=

**C**

_{i}/

**B**

_{8}and

**F**,

**H**,

**G**,

**D**,

**H′**,

**G′**are matrices defined in the Appendix A.

**B**

_{i},

**C**

_{i}coefficients enables the calculation of the pot-core sensor impedance according to the formula:

**T**,

**U**are matrices defined in the Appendix A.

## 3. Results

_{0}= R

_{0}+ jωX

_{0}in the space without conductive material was determined. In the case of the mathematical model, it was assumed that the bond coat and the substrate were nonconductive, i.e., σ

_{6}= σ

_{7}= 0. Then, the impedance of the sensor Z = R + jωX was determined after having placed it on the surface of the tested sample. All impedance measurements were performed three times, and subsequently their arithmetic mean was calculated. The values of the changes in the sensor impedance were presented as ΔZ = Z − Z

_{0}.

_{0}and reactance ΔX = X − X

_{0}obtained through the testing of both samples were normalised to reactance X

_{0}and are presented in Figure 4, Figure 5, Figure 6 and Figure 7. The measurements were made for 40 frequency values within a range of 1 kHz to 50 kHz.

## 4. Discussion

- -
- relatively low frequency in order to obtain the highest sensitivity of the sensor resistance,
- -
- high frequency in order to ensure the sensitivity of the imaginary part of the sensor impedance that is much higher than that of the real part.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**F**,

**H**,

**G**,

**D**,

**H′**,

**G′**,

**T**,

**U**were defined in the form:

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**Figure 1.**Samples for eddy current testing in the form of a thick plate made of copper (thickness 20 mm) and thin foils made of various types of conductive materials (thicknesses ranging from 0.1 mm to 0.5 mm).

**Figure 2.**Rectangular cross-section of the pot-core sensor placed above a three-layer conductive structure.

**Figure 3.**E-core sensor, I-core sensor and samples made of aluminium (σ = 36.26 MS/m), copper (σ = 58.38 MS/m), brass (σ = 17.25 MS/m) and copper (σ = 58.49 MS/m), used in the tests.

**Figure 4.**The changes in the resistance ΔR normalised with respect to the reactance X

_{0}for sample made of aluminium and copper.

**Figure 5.**The changes in the reactance ΔX normalised with respect to the reactance X

_{0}for sample made of aluminium and copper.

**Figure 6.**The changes in the resistance ΔR normalised with respect to the reactance X

_{0}for sample made of brass and copper.

**Figure 7.**The changes in the reactance ΔX normalised with respect to the reactance X

_{0}for sample made of brass and copper.

**Figure 8.**Normalised changes in the sensor resistance ΔR and reactance ΔX for thermal barrier coatings.

I-Core Sensor | E-Core Sensor | |
---|---|---|

Number of turns N | 480 | 646 |

Inner coil radius r_{1} | 2.6 mm | 4.3 mm |

Outer coil radius r_{2} | 7.8 mm | 7.3 mm |

Parameter h_{1} | 0.1 mm | 0.2 mm |

Parameter h_{2} | 15.9 mm | 3.6 mm |

Inner column radius a_{1} | 0.7 mm | 1.5 mm |

Outer column radius a_{2} | 2.5 mm | 3.7 mm |

Inner core radius c_{1} | - | 7.7 mm |

Outer core radius c_{2} | - | 9.1 mm |

Inner core height d_{1} | - | 3.7 mm |

Outer core height d_{2} | 34.5 mm | 5.3 mm |

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**MDPI and ACS Style**

Tytko, G.
Eddy Current Testing of Conductive Coatings Using a Pot-Core Sensor. *Sensors* **2023**, *23*, 1042.
https://doi.org/10.3390/s23021042

**AMA Style**

Tytko G.
Eddy Current Testing of Conductive Coatings Using a Pot-Core Sensor. *Sensors*. 2023; 23(2):1042.
https://doi.org/10.3390/s23021042

**Chicago/Turabian Style**

Tytko, Grzegorz.
2023. "Eddy Current Testing of Conductive Coatings Using a Pot-Core Sensor" *Sensors* 23, no. 2: 1042.
https://doi.org/10.3390/s23021042