# Bloch Analysis of Electromagnetic Waves in Twist-Symmetric Lines

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## Abstract

**:**

## 1. Introduction

## 2. Reducibility of Twist-Symmetric Structures

#### 2.1. Reducible and Irreducible Twist Symmetries

#### 2.2. Multimodal Transmission-Matrix Method

## 3. Twist Symmetry Conditions on a Subunit Cell

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pin-loaded coaxial transmission lines: (

**a**) 2-fold twist-symmetric, (

**b**) 4-fold twist-symmetric, (

**c**) non-twist symmetric with 2 pins, and (

**d**) non-twist symmetric with 4 pins. (${d}_{1}=2.4$ mm, ${d}_{2}=4.84$ mm, ${d}_{3}=2.4$ mm, and $h=2$ mm).

**Figure 2.**A dispersion diagram comparison of the twist-symmetric and their non-twist symmetric structures shown in Figure 1: (

**a**) with 2 pins ($p=15$ mm), (

**b**) with 2 pins ($p=7$ mm), (

**c**) with 4 pins ($p=15$ mm), and (

**d**) with 4 pins ($p=10$ mm).

**Figure 3.**A dispersion diagram of 2-fold twist-symmetric structures in Figure 1 derived from the T-matrix method and CST ES applied to their unit cells: (

**a**) Twisted $p=15$ mm, (

**b**) associated non-twisted $p=15$ mm, (

**c**) twisted $p=7$ mm, and (

**d**) associated non-twisted $p=7$ mm.

**Figure 4.**A dispersion diagram of 4-fold twist-symmetric structures in Figure 1 derived from the T-matrix method and CST ES applied to their unit cells: (

**a**) Twisted $p=15$ mm, (

**b**) associated non-twisted $p=15$ mm, (

**c**) twisted $p=10$ mm, and (

**d**) associated non-twisted $p=10$ mm.

**Figure 5.**Dispersion diagrams derived with the T-matrix method on the subunit cell of the structure in Figure 1b (p = 15 mm): (

**a**) The normalized phase constant $\beta p/\pi $ vs. frequency and (

**b**) the normalized attenuation constant $\alpha /{k}_{0}$ vs. frequency (${k}_{0}$ being the free-space wavenumber).

**Table 1.**The computational time for solving the periodic structure in Figure 1b.

1 frequency point | T matrix (subunit Cell) | T matrix (unit cell) | CST (eigensolver) |

Time (s) | 12 | 19 | 13 |

76 frequency points | T matrix (subunit cell) | T matrix (unit cell) | CST (eigensolver) |

Time (s) | 21 | 88 | 780 |

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**MDPI and ACS Style**

Bagheriasl, M.; Valerio, G.
Bloch Analysis of Electromagnetic Waves in Twist-Symmetric Lines. *Symmetry* **2019**, *11*, 620.
https://doi.org/10.3390/sym11050620

**AMA Style**

Bagheriasl M, Valerio G.
Bloch Analysis of Electromagnetic Waves in Twist-Symmetric Lines. *Symmetry*. 2019; 11(5):620.
https://doi.org/10.3390/sym11050620

**Chicago/Turabian Style**

Bagheriasl, Mohammad, and Guido Valerio.
2019. "Bloch Analysis of Electromagnetic Waves in Twist-Symmetric Lines" *Symmetry* 11, no. 5: 620.
https://doi.org/10.3390/sym11050620