# Optimization of an Inductive Displacement Transducer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{n}

^{fix}—the predefined values of the induced voltage desired to be reached once the optimization is done for each different position “n”, where the moving magnetic core is found; u

_{n}

^{c}—the values of the induced voltage in the secondary coils, calculated by the COMSOL Multiphysics 4.3 software for each “n”. In addition,

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}, p

_{6}—represent the optimization parameters.

_{0}= {15, 20, 2.25, 4.5, 4.5, 26.50}

^{T}(mm)

^{−6}(m

^{2}). The dimensions of the geometry used are detailed in Table 1 together with the value of the objective function for the initial configuration of the test model.

#### 2.1. The First Approach

#### 2.2. The Second Approach—Extending the Operational Range from 0 to 18 (mm)

## 3. Results

#### 3.1. NSGA-II Optimization

#### 3.2. NSGA-III Optimization

## 4. Discussion

#### 4.1. The Discussion Based on the NSGA-II Optimization

#### 4.2. The Discussion Based on the NSGA-III Optimization

#### 4.3. Results Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The graphical representation of the calibration between initial model and test model regarding the output voltage. The green and red lines represent the Reference Data results outlined in [3].

**Figure 4.**Graphical representation of the optimization. Comparison between the initial and the test models. The green line represents the Reference Data results outlined in [3].

**Figure 6.**The graphical representation of the values of the LVDT with the extension of the work area up to 18 (mm). The blue line represents the Reference Data results outlined in [3].

**Figure 7.**The graphical representation of the output voltage for the NSGA-II optimized model with the expansion of the linage up to 18 (mm). The blue line represents the Reference Data results outlined in [3].

**Figure 8.**The graphical representation of the output voltage for optimized model with the NSGA-II algorithm in comparison with the imposed output.

**Figure 9.**The graphical representation of the magnetic flux density for the test model optimized with the NSGA-II algorithm.

**Figure 10.**The graphical representation of the output voltage NSGA-III optimized model with the expansion of the work area up to 18 (mm). The blue line represents the Reference Data results outlined in [3].

**Figure 11.**The graphical representation of the output voltage for the optimized model with the NSGA-III algorithm in comparison with the imposed output.

**Figure 12.**The graphical representation of the magnetic flux density for the test model optimized with the NSGA-III algorithm.

**Figure 13.**The graphical representation of the convergence of the objective function while using the NSGA-II algorithm.

**Figure 14.**The graphical representation of the best individuals that resulted from the use of the NSGA-II algorithm.

**Figure 15.**The graphical representation of the convergence of the objective function while using the NSGA-III algorithm.

**Figure 16.**The graphical representation of the best individuals that resulted from the use of the NSGA-III algorithm.

**Figure 17.**The graphical representation of the comparison between optimization results made with NSGA-II and NSGA-III.

**Figure 18.**The graphical representation of the comparison between optimization results obtained with NSGA-II and NSGA-III regarding the imposed output.

**Figure 19.**(

**a**) A two-dimensional section of the model’s details compared to the simulation model configurations (blue), the model optimized with NSGA-II (red), and the model optimized with NSGA-III (green). The configurations of the model optimized with NSGA-II in 3D (

**b**) and the model optimized with NSGA-III in 3D (

**c**).

Initial Model | Value |
---|---|

Length of the secondary coil—p1 | 15 (mm) |

Length of the magnetic core—p2 | 20 (mm) |

Width of the magnetic core—p3 | 2.25 (mm) |

Thickness of the casing—p4 | 4.50 (mm) |

Distance casing–secondary coil—p5 | 4.50 (mm) |

Width of the casing—p6 | 26.50 (mm) |

Standardized position of the magnetic core—p7 | 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 (mm) |

Length of the secondary coils that maintains the same values—p8 | 15 (mm) |

Length of the primary coil that maintains the same values—p9 | 20 (mm) |

Height of the LVDT that maintains the same values—p10 | 74 (mm) |

Radius of the LVDT from the center of the magnetic core, at shell extremity, that maintains the same values—p11 | 32.50 (mm) |

Value of the objective function—f(p_{0}) | 1.7334 × 10^{−8} (V^{2}) |

Parameter 2D Cross Section NSGA-II | Value |
---|---|

Length of the secondary coil | 7.69 (mm) |

Length of the magnetic core | 14.87 (mm) |

Width of the magnetic core | 4.90 (mm) |

Width of the casing | 31.24 (mm) |

Thickness of the casing | 4.76 (mm) |

Distance casing–secondary coil | 1.61 (mm) |

Standardized position of the magnetic core | 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 (mm) |

Value of the objective function | 2.37856 × 10^{−10} (V^{2}) |

Parameter 2D Cross Section NSGA-III | Value |
---|---|

Length of the secondary coil | 8.06 (mm) |

Length of the magnetic core | 16.56 (mm) |

Width of the magnetic core | 5.53 (mm) |

Width of the casing | 30.76 (mm) |

Thickness of the casing | 1.62 (mm) |

Distance casing–secondary coil | 2.62 (mm) |

Standardized position of the magnetic core | 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 (mm) |

Value of the objective function | 2.29 × 10^{−9} (V^{2}) |

(mm) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
---|---|---|---|---|---|---|---|---|---|---|

Linear val (mV) | 0 | 0.1164 | 0.2329 | 0.3493 | 0.4658 | 0.5822 | 0.6987 | 0.8151 | 0.9316 | 1.0479 |

Original output in [3] (mV) | 0 | 0.1164 | 0.225 | 0.3215 | 0.404 | 0.4673 | No data | No data | No data | No data |

Optimal output in [3] (mV) | 0 | 0.1183 | 0.2375 | 0.3555 | 0.4679 | 0.5596 | No data | No data | No data | No data |

Obtain val (mV) | 0 | 0.01167 | 0.2264 | 0.3233 | 0.4041 | 0.4656 | 0.5056 | 0.5218 | 0.5152 | 0.4847 |

NSGA-II val (mV) | 0 | 0.1139 | 0.2299 | 0.3467 | 0.4675 | 0.5866 | 0.7039 | 0.8188 | 0.9281 | 1.0361 |

NSGA-III val (mV) | 0 | 0.1160 | 0.2354 | 0.3549 | 0.4770 | 0.5968 | 0.7089 | 0.8171 | 0.9191 | 1.0074 |

Initial rel. er (%) | 0 | 0.2975 | 2.8902 | 8.0255 | 15.268 | 25.049 | 38.187 | 56.225 | 80.796 | 116.23 |

Rel.er. NSGA-II (%) | 0 | 2.2208 | 1.2641 | 0.7458 | 0.3620 | 0.7536 | 0.7491 | 0.4509 | 0.3721 | 1.1442 |

Rel.er. NSGA-III (%) | 0 | 0.3270 | 1.0766 | 1.5950 | 2.3495 | 2.4415 | 1.4515 | 0.2427 | 1.3585 | 4.0294 |

Parameters | Initial Model | NSGA-II | NSGA-III |
---|---|---|---|

p₁ (mm) | 15 | 7.69 | 8.06 |

p₂ (mm) | 20 | 14.87 | 16.56 |

p₃ (mm) | 2.25 | 4.90 | 5.53 |

p₄ (mm) | 4.50 | 31.24 | 30.76 |

p₅ (mm) | 4.50 | 4.76 | 1.62 |

p₆ (mm) | 26.50 | 1.61 | 2.62 |

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**MDPI and ACS Style**

Mociran, B.; Gliga, M.
Optimization of an Inductive Displacement Transducer. *Sensors* **2023**, *23*, 8152.
https://doi.org/10.3390/s23198152

**AMA Style**

Mociran B, Gliga M.
Optimization of an Inductive Displacement Transducer. *Sensors*. 2023; 23(19):8152.
https://doi.org/10.3390/s23198152

**Chicago/Turabian Style**

Mociran, Bogdan, and Marian Gliga.
2023. "Optimization of an Inductive Displacement Transducer" *Sensors* 23, no. 19: 8152.
https://doi.org/10.3390/s23198152