# Comparative Study of Machine-Learning Frameworks for the Elaboration of Feed-Forward Neural Networks by Varying the Complexity of Impedimetric Datasets Synthesized Using Eddy Current Sensors for the Characterization of Bi-Metallic Coins

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works in Machine-Learning

#### 1.2. Related Works in Eddy Current Sensors

#### 1.3. Methodology

## 2. Principle of Eddy Current Sensor and Application in Coin Classification

#### 2.1. Eddy Current Measurements on Bi-Metallic Coins

#### 2.2. Analytical Modelling and Parameter Extraction

_{0}: Permeability of air, I: Excitation current, r

_{0}: Average radius of the coil, J

_{1}: First order Bessel function, α: Separation constant, l: Distance of the center of the coil to the coin surface, c: Thickness of the top layer of the coin, ω: Angular frequency, i: integer ‘0-2’, µ

_{1}: Permeability of the top layer of the coin, µ

_{2}: Permeability of the center layer of the coin, ∈

_{0}: Permittivity of the air, ∈

_{1}: Permittivity of the top layer of the coin, ∈

_{2}: Permittivity of the center layer of the coin, σ

_{0}: Conductivity of the air, σ

_{1}: Conductivity of the top layer of the coin, σ

_{2}: Conductivity of the center layer of the coin, L: Inductance of the sensor coil, N: Number of turns of the coil

_{Coin}) from inner inductance inside the wire (∆L

_{Air}) due to skin effect, which is a coil in the air without target material [20,21]. The measured spectra and synthesized spectra using the analytical model are identical and fit to each other over the entire frequency range.

#### 2.3. Interpolation Technique to Generate Datasets with Different Challenging Levels

## 3. Interpretation of Synthesized Data

^{2}(chi

^{2}) is used [22]. The calculated scores for all normalized six features show that the phase at 10 kHz and 40 kHz contributes the most in classification as can be seen in Table 1.

## 4. Implementation and Training of Neural Network Frameworks

## 5. Results

#### 5.1. Model Size

#### 5.2. Model Training Time

#### 5.3. Prediction Time

#### 5.4. Prediction Accuracy

#### 5.4.1. NN with 8 Neurons in Each Hidden Layer

#### 5.4.2. NN with 16 Neurons in Each Hidden Layer

#### 5.4.3. NN with 32 Neurons in Each Hidden Layer

#### 5.4.4. NN with 64 Neurons in Each Hidden Layer

#### 5.5. Framework Evaluation

#### 5.6. Reliability Evaluation of Selected Framework

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Methodology of Investigations for Four Frameworks ‘Keras with TensorFlow, Pytorch, TensorFlow, and CNTK’.

**Figure 2.**Representation of Eddy Current Sensor Coil over Bi-Metallic Coin (

**a**) Schematic of Measurement Set-Up. (

**b**) Side View of Magnetic Field Excitation at Different Frequencies.

**Figure 3.**Measured Inductance Spectra of EUR 2 Coin using Reverse and Obverse Side at different Rotating Angles from 0° to 360° Spanned each at 45°.

**Figure 4.**(

**a**) Real Value of Measured and Synthesized Intermediate Inductance Spectra for EUR 2 and TRY 1 Coin. (

**b**) Imaginary Value of Measured and Synthesized Intermediate Inductance Spectra for EUR 2 and TRY 1 Coin.

**Figure 5.**Real Values of Synthesized Intermediate Inductance Spectra at different Difficulty Levels of TRY 1 Coin by Bringing the Parameters of Coin Closer to EUR 2 Coin.

**Figure 6.**Inductance Ratio vs. Inductance Phase for EUR 2 and TRY 1 Coin Classes at Three Frequencies Measured 10,000 Times Each.

**Figure 7.**Structure of NN Implemented for 6 Features and 1 Output with 2 Hidden Layers Each Containing 8 Number of Neurons.

**Figure 9.**Model Training Time for Different Frameworks using Different Number of Neurons in Hidden Layers.

**Figure 10.**Prediction Time for Different Frameworks using Different Number of Neurons in Hidden Layers.

**Figure 11.**Prediction Accuracy at Different Closing Distances between TRY 1 and EUR 2 Coins using Different Number of Neurons in Hidden Layers for Different Frameworks: (

**a**) Keras with TensorFlow Backend (

**b**) Pytorch (

**c**) TensorFlow (

**d**) CNTK.

**Figure 12.**Comparison of the Performance Metrics considering Model Size, Prediction Time and Model Training Time for Pytorch and TensorFlow with 32 and 64 Neurons in each Hidden Layers.

**Figure 13.**Reliability Evaluation of Pytorch with 32 Neurons in each Hidden Layer by Calculating the Mean Value, Minimum and Maximum Deviation using 10 Measurements for Accuracy Metric.

**Table 1.**Features Score Calculated using χ2 Test (Higher the score is more contribution of a feature during classification).

Frequency | Score | |
---|---|---|

Impedance | Phase | |

10 kHz | 0.00146 | 6.346198 |

40 kHz | 0.000571 | 12.16268 |

1 MHz | 0.000091 | 0.865512 |

Hidden Layers | Neurons in Each Layer | Activation Function | Dropout Layers | Dropout Ratio | Activation Function (Output Layer) | No. of Epochs | Optimizer |
---|---|---|---|---|---|---|---|

2 | 8–64 | ReLU (Rectified Linear Unit) | 2 | 0.3 | Sigmoid | 10, *100 in CNTK | Adam |

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**MDPI and ACS Style**

Munjal, R.; Arif, S.; Wendler, F.; Kanoun, O.
Comparative Study of Machine-Learning Frameworks for the Elaboration of Feed-Forward Neural Networks by Varying the Complexity of Impedimetric Datasets Synthesized Using Eddy Current Sensors for the Characterization of Bi-Metallic Coins. *Sensors* **2022**, *22*, 1312.
https://doi.org/10.3390/s22041312

**AMA Style**

Munjal R, Arif S, Wendler F, Kanoun O.
Comparative Study of Machine-Learning Frameworks for the Elaboration of Feed-Forward Neural Networks by Varying the Complexity of Impedimetric Datasets Synthesized Using Eddy Current Sensors for the Characterization of Bi-Metallic Coins. *Sensors*. 2022; 22(4):1312.
https://doi.org/10.3390/s22041312

**Chicago/Turabian Style**

Munjal, Rohan, Sohaib Arif, Frank Wendler, and Olfa Kanoun.
2022. "Comparative Study of Machine-Learning Frameworks for the Elaboration of Feed-Forward Neural Networks by Varying the Complexity of Impedimetric Datasets Synthesized Using Eddy Current Sensors for the Characterization of Bi-Metallic Coins" *Sensors* 22, no. 4: 1312.
https://doi.org/10.3390/s22041312