^{1,a)}, F. Calvano

^{2}, C. Nappi

^{1}and E. Sarnelli

^{1}

### Abstract

We studied the eddy currents excited by a time varying external magnetic field in thin metallic plates in the presence of a circular hole piercing the plate. The value of the normal component of the magnetic field over the circular defect is analytically calculated and a complete scanning magnetic operation along a line crossing the defect is simulated. The analytical solution is then tested against a direct numerical simulation with good results. The aim is the reconstruction and interpretation of magnetic signatures due to structural defects in nondestructive evaluation made by superconducting quantum interference devicemicroscopymeasurements.

I. INTRODUCTION

II. EDDY CURRENTS IN THIN CONDUCTING PLATES

III. EDDY CURRENTS IN THE ABSENCE OF FLAWS

IV. DEVIATION FROM AXIAL SYMMETRY: A FLAWED PLATE

A. Magnetic field distribution

B. Eddycurrent density in the plate

C. Magnetic field at the sensor location

V. RESULTS AND DISCUSSION

A. Comparison with direct numerical simulations

VI. CONCLUSIONS

### Key Topics

- Magnetic fields
- 31.0
- Eddies
- 27.0
- Magnetic field sensors
- 18.0
- Superconducting quantum interference devices
- 13.0
- Current density
- 9.0

## Figures

Schematic representation of the simulated scanning operation over a flawed metallic plate. The midplane of the plate coincides with the plane.

Schematic representation of the simulated scanning operation over a flawed metallic plate. The midplane of the plate coincides with the plane.

The -component of the magnetic field generated by a circular coil of radius at different distances from the coil plane as a function of the radial coordinate . The gray lines illustrate the step function used in this work to approximate the field generated by the coil. For simplicity, we choose . The field is normalized to its value in the center of the coil, .

The -component of the magnetic field generated by a circular coil of radius at different distances from the coil plane as a function of the radial coordinate . The gray lines illustrate the step function used in this work to approximate the field generated by the coil. For simplicity, we choose . The field is normalized to its value in the center of the coil, .

Comparison between the solution for the distribution of the eddy current found in this work (solid line) for the unflawed plate and the Dodd and Deeds theory (Ref. 4) (dashed line). Note the linear growth of the current intensity between and

Comparison between the solution for the distribution of the eddy current found in this work (solid line) for the unflawed plate and the Dodd and Deeds theory (Ref. 4) (dashed line). Note the linear growth of the current intensity between and

Current lines in the presence of a circular hole [centered in (0, 0)] calculated as level curves of the magnetic field function derived in this work (arbitrary units). The outer circle sets the extension of the eddy currents.

Current lines in the presence of a circular hole [centered in (0, 0)] calculated as level curves of the magnetic field function derived in this work (arbitrary units). The outer circle sets the extension of the eddy currents.

Simulated magnetic field profiles for different sensor lift-off values: (a) 5 mm radius hole, (b) 2 mm radius hole. The other parameters used for the simulations are: , , , and . Starting from the top to the bottom curve the scanning distance, in both figures, increase as follows: .

Simulated magnetic field profiles for different sensor lift-off values: (a) 5 mm radius hole, (b) 2 mm radius hole. The other parameters used for the simulations are: , , , and . Starting from the top to the bottom curve the scanning distance, in both figures, increase as follows: .

Comparison between the magnetic field profiles obtained by our analytical model (solid line) and direct numerical simulations (open dots): (a) 5 mm radius hole and (b) 2 mm radius hole. For the comparison, we have chosen the first four curves of Fig. 5 corresponding to scanning heights , respectively.

Comparison between the magnetic field profiles obtained by our analytical model (solid line) and direct numerical simulations (open dots): (a) 5 mm radius hole and (b) 2 mm radius hole. For the comparison, we have chosen the first four curves of Fig. 5 corresponding to scanning heights , respectively.

Schematic representation of the simulated scanning. The wider circle of radius represents the plate zone in which eddy currents are confined. As this zone moves with the coil (and the sensor), from the left to the right, over the hole, the -component of the magnetic field is calculated at the sensor position (the center of the wider circle ) by an integration over the regions denoted as 1, 2, 3, and 4. During this process, the hole boundary (the smaller circle of radius ) remains confined within the circle of radius , i.e., within the zone of the eddy current extension.

Schematic representation of the simulated scanning. The wider circle of radius represents the plate zone in which eddy currents are confined. As this zone moves with the coil (and the sensor), from the left to the right, over the hole, the -component of the magnetic field is calculated at the sensor position (the center of the wider circle ) by an integration over the regions denoted as 1, 2, 3, and 4. During this process, the hole boundary (the smaller circle of radius ) remains confined within the circle of radius , i.e., within the zone of the eddy current extension.

## Tables

Values of the external magnetic field (nT) at the midplane of the plate ( plane) generated in the center coil for a given lift-off . It is also reported the calculated background signal [Eq. (A3)] for the case of Fig. 5.

Values of the external magnetic field (nT) at the midplane of the plate ( plane) generated in the center coil for a given lift-off . It is also reported the calculated background signal [Eq. (A3)] for the case of Fig. 5.

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