# Analysis of Periodic Structures Made of Pins Inside a Parallel Plate Waveguide

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## Abstract

**:**

## 1. Introduction

## 2. Geometries Considered in the Analysis

## 3. Stopband Analysis

#### Parametric Study for Cases with Pins of Unequal Heights

## 4. Equivalent Refractive Index

#### Effect of the Different Parameters on the Equivalent Refractive Index

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CST | Computer Simulation Studio |

PPWG | Parallel plate waveguide |

AMC | Artificial magnetic conductor |

TEM | Transverse electromagnetic mode |

## References

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**Figure 1.**Pin geometries considered in the study. * In these cases, we will consider two cases of shift in one direction (X) and two directions (XY) shifts.

**Figure 2.**Unit cell description including axes. Top view of the translations in only X (

**b**) or XY (

**c**) directions.

**Figure 3.**Dispersion diagrams for cases a, b and c according to Figure 1. The frequency is normalized to the frequency corresponding to the single-pin height ${h}_{pin}=\lambda /4$. Blue lines correspond to the first mode and red lines to the second mode.

**Figure 4.**Dispersion diagrams for case d in Figure 1 for different pin heights, where ${h}_{top}+{h}_{bottom}={h}_{pin}$. Blue lines represent the first mode and red lines the second mode.

**Figure 5.**Dispersion diagrams for case e in Figure 1 for different pin heights, where ${h}_{top}+{h}_{bottom}={h}_{pin}$. Blue lines represent the first mode and red lines the second mode.

**Figure 6.**Dispersion diagrams for pins in case f according to Figure 1. Different colors represent different modes.

**Figure 7.**Parametric study for the geometry described as d in Figure 1 for h

_{bottom}= 1.5 mm. Red lines represent the start frequency of the stopband whilst blue lines represent its end frequency.

**Figure 8.**Parametric study for the geometry described as d in Figure 1 for h

_{bottom}= 2.0 mm. Red lines represent the start frequency of the stopband whilst blue lines represent its end frequency.

**Figure 9.**Parametric study for the X-shifted geometry with h

_{top}= h

_{bottom}, p

_{1}= 0.125λ

_{0}, p

_{2}= 0.1875λ

_{0}, p

_{3}= 0.25λ

_{0}, p

_{4}= 0.375λ

_{0}, p

_{5}= 0.5λ

_{0}, g

_{1}= 0.0125λ

_{0}, g

_{2}= 0.0625λ

_{0}, and g

_{3}= 0.125λ

_{0}.

**Figure 10.**Parametric study for the X-shifted geometry with h

_{bottom}= 2 mm, p

_{1}= 0.125λ

_{0}, p

_{2}= 0.1875λ

_{0}, p

_{3}= 0.25λ

_{0}, p

_{4}= 0.375λ

_{0}, p

_{5}= 0.5λ

_{0}, g

_{1}= 0.0125λ

_{0}, g

_{2}= 0.0625λ

_{0}, and g

_{3}= 0.125λ

_{0}.

**Figure 11.**Parametric study for the XY-shifted geometry with h

_{top}= h

_{bottom}, p

_{1}= 0.125λ

_{0}, p

_{2}= 0.1875λ

_{0}, p

_{3}= 0.25λ

_{0}, p

_{4}= 0.375λ

_{0}, p

_{5}= 0.5λ

_{0}, g

_{1}= 0.0125λ

_{0}, g

_{2}= 0.0625λ

_{0}, and g

_{3}= 0.125λ

_{0}.

**Figure 12.**Parametric study for the XY-shifted geometry with h

_{bottom}= 2 mm, p

_{1}= 0.125λ

_{0}, p

_{2}= 0.1875λ

_{0}, p

_{3}= 0.25λ

_{0}, p

_{4}= 0.375λ

_{0}, p

_{5}= 0.5λ

_{0}, g

_{1}= 0.0125λ

_{0}, g

_{2}= 0.0625λ

_{0}, and g

_{3}= 0.125λ

_{0}.

**Figure 13.**Equivalent refractive indices for different cases from Figure 1: in particular cases, a, b, d and f.

**Figure 14.**Equivalent refractive indices for different cases in Figure 1 after shifting the pins.

**Figure 15.**Effect of varying different parameters in the geometries a, b and d from Figure 1. The reference case has a gap of 0.5 mm, a period of 2 mm and a width of 1 mm.

**Figure 16.**Effect of varying different parameters in the geometry f (interleaved pins) from Figure 1. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.

**Figure 17.**Effect of varying different parameters in the geometry c from Figure 1. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.

**Figure 18.**Effect of varying different parameters in the geometry e from Figure 1. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.

**Table 1.**Summary of the starting and end frequencies of the stopbands created by the different analyzed structures.

Geometry | ${\mathit{f}}_{\mathit{start}}$ | ${\mathit{f}}_{\mathit{end}}$ |
---|---|---|

Single pin (case a) | 0.84${f}_{0}$ | 1.5${f}_{0}$ |

Half-pin (case b) | 1.28${f}_{0}$ | 1.7${f}_{0}$ |

Pins, different heights (case d, ${h}_{1}$ = 1.5) | 1.23${f}_{0}$ | 1.68${f}_{0}$ |

Pins, different heights (case d, ${h}_{1}$ = 2) | 1.01${f}_{0}$ | 1.58${f}_{0}$ |

Pins, shifted (case e, ${h}_{1}$ = 1.5) | 1.29${f}_{0}$ | 1.51${f}_{0}$ |

Pins, shifted (case e, ${h}_{1}$ = 2) | 1.03${f}_{0}$ | 1.55${f}_{0}$ |

Pins, glide (case e, ${h}_{1}$ = 1.5) | 1.31${f}_{0}$ | 1.53${f}_{0}$ |

Pins, glide (case e, ${h}_{1}$ = 2) | 1.04${f}_{0}$ | 1.55${f}_{0}$ |

Interleaved pins (case f, ${h}_{1}$ = 1.5) | 1.31${f}_{0}$ | 1.68${f}_{0}$ |

Interleaved pins (case f, ${h}_{1}$ = 2) | 1.38${f}_{0}$ | 1.57${f}_{0}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Memeletzoglou, N.; Sanchez-Cabello, C.; Pizarro-Torres, F.; Rajo-Iglesias, E.
Analysis of Periodic Structures Made of Pins Inside a Parallel Plate Waveguide. *Symmetry* **2019**, *11*, 582.
https://doi.org/10.3390/sym11040582

**AMA Style**

Memeletzoglou N, Sanchez-Cabello C, Pizarro-Torres F, Rajo-Iglesias E.
Analysis of Periodic Structures Made of Pins Inside a Parallel Plate Waveguide. *Symmetry*. 2019; 11(4):582.
https://doi.org/10.3390/sym11040582

**Chicago/Turabian Style**

Memeletzoglou, Nafsika, Carlos Sanchez-Cabello, Francisco Pizarro-Torres, and Eva Rajo-Iglesias.
2019. "Analysis of Periodic Structures Made of Pins Inside a Parallel Plate Waveguide" *Symmetry* 11, no. 4: 582.
https://doi.org/10.3390/sym11040582