# A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### A − φ Formulation in Multiconductor Winding Structures

## 3. Numerical Implementation of the Derived $\mathbf{A}-\mathit{\phi}$ Formulation on a 2D Cross-Section of an MFT

- An FEM model was implemented across all regions of the winding window, which was named ${\Omega}_{\mathrm{FEM}}$. This model compares triangular and rectilinear mesh types with a focus on the computational efficiency and accuracy (see Figure 2a,b).
- A hybrid FEM–SEM model was developed, ${\Omega}_{\mathrm{SEM}}$, including the clearance distances, where the SEM was coupled with the FEM model employed on the winding regions (see Figure 2c).

**Figure 1.**(

**a**) The MFT under-study, (

**b**) the cross-section area of the MFT winding, (

**c**) the graphical representation of applied FEM-SEM.

**Figure 2.**Meshes generated using (

**a**) triangular elements and (

**b**) rectilinear elements; (

**c**) a hybrid FEM–SEM model. All three discretizations exhibit equal number of elements for the geometry depicted in Figure 1.

#### 3.1. Solution of MQS Formulation Using FEM

- The current flowing through the winding $\mathrm{j}$ is uniform:${i}_{n}-{i}_{\mathrm{s},\phantom{\rule{3.33333pt}{0ex}}\mathrm{j}}=0$ for every conductor n belonging to the winding $\mathrm{j}$, where ${i}_{\mathrm{s},\phantom{\rule{3.33333pt}{0ex}}\mathrm{j}}$ is the source current at the terminal of winding $\mathrm{j}$.
- The voltage drops of the conductors, ${v}_{n}$, belonging to the winding $\mathrm{j}$ add up to the terminal voltage drop of the winding $\mathrm{j}$, ${V}_{\mathrm{s},\mathrm{j}}$:$${V}_{\mathrm{s},\mathrm{j}}-\sum {v}_{n}=0.$$

#### 3.2. Spectral Element Method Formulation

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AC | Alternating Current |

dof | Degrees of Freedom |

FEM | Finite Element Method |

SEM | Spectral Element Method |

MFT | Medium-Frequency Transformers |

MQS | Magnetoquasistatic |

1D, 2D, 3D | One-, Two-, Three-dimensional |

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**Figure 3.**The AC resistance R and reactance X of (

**a**) MFT 1 (

**b**) MFT 2, and (

**c**) MFT 3 computed using extra-fine, fine, and coarse mesh elements with triangular and rectilinear shapes, as well as the hybrid FEM–SEM method and the corresponding deviations from the extra-fine reference model.

**Figure 4.**The distribution of ${A}_{\mathrm{z}}$ at the frequency of 50 kHz, which was solved using (

**a**) FEM employing a rectangular mesh (${A}_{\mathrm{z}}^{\mathrm{FEM}-\mathrm{Recti}}$), (

**b**) the pointwise deviation of coupled FEM–SEM method (${A}_{\mathrm{z}}^{\mathrm{FEM}-\mathrm{SEM}}$), and (

**c**) with FEM utilizing a triangular mesh (${A}_{\mathrm{z}}^{\mathrm{FEM}-\mathrm{Tri}}$); both compared with ${A}_{\mathrm{z}}^{\mathrm{FEM}-\mathrm{Recti}}$.

**Figure 5.**The comparison of computational costs for FEM with triangular and rectangular meshes, as well as the hybrid FEM–SEM model, solved for the test bench MFT 1 at the frequency of 20 kHz.

Dimensions | Symbol | MFT 1 | MFT 2 | MFT 3 |
---|---|---|---|---|

Mumber of winding turns | ${N}_{1}/{N}_{2}$ | 10/10 | 10/10 | 10/10 |

Foil thickness | ${d}_{\mathrm{w}}$ | 1 [mm] | 0.2 [mm] | 1 [mm] |

Foil height | ${h}_{\mathrm{w}}$ | 100 [mm] | 100 [mm] | 100 [mm] |

Interlayer insulation thickness | ${d}_{\mathrm{ins}}$ | 0.2 [mm] | 0.2 [mm] | 0.2 [mm] |

Windings–core clearance distance | D | 20 [mm] | 20 [mm] | 50 [mm] |

Core window height | ${h}_{\mathrm{c}}$ | 140 [mm] | 140 [mm] | 200 [mm] |

Mesh Type | Extra-Fine | Fine | Coarse |
---|---|---|---|

Triangular | - | 21,872 | 8310 |

Rectilinear | 28,444 | 15,795 | 7422 |

FEM–SEM | - | 6950 | 5478 |

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## Share and Cite

**MDPI and ACS Style**

Pourkeivannour, S.; van Zwieten, J.S.B.; Friedrich, L.A.J.; Curti, M.; Lomonova, E.A.
A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings. *J* **2023**, *6*, 627-638.
https://doi.org/10.3390/j6040041

**AMA Style**

Pourkeivannour S, van Zwieten JSB, Friedrich LAJ, Curti M, Lomonova EA.
A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings. *J*. 2023; 6(4):627-638.
https://doi.org/10.3390/j6040041

**Chicago/Turabian Style**

Pourkeivannour, Siamak, Joost S. B. van Zwieten, Léo A. J. Friedrich, Mitrofan Curti, and Elena A. Lomonova.
2023. "A Comparative Study of Finite Element Method and Hybrid Finite Element Method–Spectral Element Method Approaches Applied to Medium-Frequency Transformers with Foil Windings" *J* 6, no. 4: 627-638.
https://doi.org/10.3390/j6040041