# A Subcell Finite-Difference Time-Domain Implementation for Narrow Slots on Conductive Panels

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## Abstract

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## 1. Introduction

- The finite-difference time-domain (FDTD) method typically uses a mesh size smaller than one-tenth of the wavelength, $\lambda >10\Delta $; therefore, the width of a narrow slot is assumed to be $w\ll \Delta <\lambda $ (where $\lambda $ is the wavelength, $\Delta $ is the cell size of the mesh, and w is the width of the slot). Consequently, DMMA [7] is inherently applicable only when assuming a uniform field distribution on the slot;
- It assumes waves are perpendicularly impinging on the slot, with the electric field polarized in the direction across it;
- The magnetic condition must be placed on the dual grid, not matching the primary grid alignment used for PEC and dielectrics, thus making it tricky to implement in FDTD meshers.

## 2. Narrow Slots in FDTD

#### 2.1. DMMA Model

#### 2.2. Conformal Approximation (CA) Model

#### 2.3. Subgridding Approximation Model

## 3. CFC Numerical Modeling: SGBC for Lossy Thin Panels

## 4. Results

- A.
- A classical validation consisting of a narrow slot placed on an indefinite PEC or CFC, aiming to examine the limits of the normal propagation hypothesis and the field homogeneity along the slot when the thickness is the same as the FDTD cell size;
- B.
- A typical PEC cage with one of its sides covered either by PEC or CFC, including a rectangular slot with the same width as the FDTD cell size. We evaluate the SE at the center of the cage when a tilted plane wave illuminates it. Here, we seek to assess the actual robustness of DMMA in a highly resonant scenario;
- C.
- Finally, a real PEC cage with one side, either in PEC or CFC, is tested in an RC, and experimental results are known. To mimic the experimental RC in FDTD simulations, we employ a stochastic plane-wave incidence [25]. Results aim to show a typical EMC real scenario to illustrate the expected differences in simulation versus measurements.

#### 4.1. Slot on Indefinite Plate

- The effects of non-normal propagation are relevant: we illuminate one side of the slot with a dipole source and observe the field on the other side at a point where there is no straight line of vision to the source (Figure 5 shows the positions of the dipole and probe with respect to the slot);
- The slot width is not thin with respect to the FDTD space step, specifically $w=\Delta $.

#### 4.2. Enclosure under Plane Wave Incidence

#### 4.3. Cage with Curved Slots: Numerical and Experimental Data

## 5. Conclusions

- A classical validation consisting of a narrow slot on PEC and conductive panel, where DMMA assumptions of the normal propagation and uniform field distribution across it produce worse results than CA at HF;
- A PEC cage with one of its sides covered either by PEC or conductive panel with a rectangular slot under plane wave incidence. There, the actual robustness of DMMA is shown compared to the full-wave CA method since the cage resonances dominate over the slot ones at HF;
- A real PEC cage with one side, either in PEC or in CFC, in a RC compared with experimental results. Results show a typical EMC real scenario and help the engineer understand the margins in the differences in simulation versus measurements. Results show that DMMA underestimates the SE, especially below the first resonance. This effect does not appear in the well-controlled cage of the previous test case, and we have attributed it to the stochastic nature of the incident waves. For higher frequencies, no clear conclusions can be drawn since the cage resonances clutter the results, and only worst-case margins could be extracted by an EMC practitioner.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FDTD | Finite-difference time-domain |

EMC | Electromagnetic compatibility |

EMI | Electromagnetic interference |

LIE | Lightning indirect effects |

CFC | Carbon Fiber Composites |

CFL | Courant–Friedrichs–Lewy |

DMMA | Dispersive magnetic material approximation |

PEC | Perfect electrically conducting |

CA | Conformal approximation |

SG | Subgridding |

RC | Reverberating chamber |

SE | Shielding effectiveness |

IL | Insertion loss |

HF | High frequencies |

MF | Medium frequencies |

LF | Low frequencies |

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**Figure 2.**FDTD cells with a narrow slot centered on the primary grid (electric edges). Furthermore, details of the fields that are involved in the CA model.

**Figure 3.**Cross-section of a conformal cell used for aperture treatment using the CA procedure. Detail of the electric (E) and magnetic (H) fields involved in the method.

**Figure 4.**Cross-section of a subgridded region with a refinement ratio of 1:2 and 3 subgrid divisions, ${N}_{\mathrm{sg}}=3$. Each color corresponds to cells at the same subgrid level. The aperture of the slot is located in the finest grid.

**Figure 6.**Comparison of the IL results of CA and DMMA methods for a narrow slot on a conductive indefinite plane.

**Figure 7.**Geometry of a rectangular PEC cavity with slot mounted on conductive face. The slot dimensions are 100 mm × 5 mm.

**Figure 8.**Electric field shielding effectiveness (${\mathrm{SE}}_{\mathrm{ee}}$) evaluated at the central position of the rectangular enclosure. These results correspond to the Test Case shown in Figure 7.

**Figure 9.**Magnetic field shielding effectiveness (${\mathrm{SE}}_{\mathrm{hh}}$) evaluated at the central position of the rectangular enclosure. These results correspond to the Test Case shown in Figure 7.

**Figure 10.**Experimental setup of a rectangular enclosure with a semicircular slot on the front panel.

Method | CFLN | ${\mathbf{\Delta}}_{\mathbf{min}}$ [mm] | ${\mathbf{\Delta}}_{\mathbf{max}}$ [mm] | CPU Gain Respect to Standard FDTD | Memory Size [GB] |
---|---|---|---|---|---|

Standard FDTD | 0.9 | 1.25 | - | 1.0 | 7.9 |

CA | 0.9 | 5 | - | 262 | 0.1 |

DMMA | 0.9 | 5 | - | 262 | 0.1 |

SG ${N}_{\mathrm{sg}}$ = 1 | 0.67 | 1.25 | 2.5 | 4.2 | 0.68 |

SG ${N}_{\mathrm{sg}}$ = 3 | 0.67 | 1.25 | 10 | 14 | 0.5 |

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## Share and Cite

**MDPI and ACS Style**

Ruiz Cabello, M.; Martín Valverde, A.J.; Plaza, B.; Frövel, M.; Poyatos, D.; R. Bretones, A.; G. Bravo, A.; G. García, S.
A Subcell Finite-Difference Time-Domain Implementation for Narrow Slots on Conductive Panels. *Appl. Sci.* **2023**, *13*, 8949.
https://doi.org/10.3390/app13158949

**AMA Style**

Ruiz Cabello M, Martín Valverde AJ, Plaza B, Frövel M, Poyatos D, R. Bretones A, G. Bravo A, G. García S.
A Subcell Finite-Difference Time-Domain Implementation for Narrow Slots on Conductive Panels. *Applied Sciences*. 2023; 13(15):8949.
https://doi.org/10.3390/app13158949

**Chicago/Turabian Style**

Ruiz Cabello, Miguel, Antonio J. Martín Valverde, Borja Plaza, Malte Frövel, David Poyatos, Amelia R. Bretones, Alberto G. Bravo, and Salvador G. García.
2023. "A Subcell Finite-Difference Time-Domain Implementation for Narrow Slots on Conductive Panels" *Applied Sciences* 13, no. 15: 8949.
https://doi.org/10.3390/app13158949