# Twist and Glide Symmetries for Helix Antenna Design and Miniaturization

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Baseline Helix Antenna Designs

_{o}, the helix circumference is considerably smaller than the wavelength (2πr << λ

_{o}) [17]. The polarization in this mode is typically aimed to be circular. The antenna operates in its axial mode if the helix circumference is in the order of the wavelength (2πr ≈ λ

_{o}) and the pitch distance p is a quarter of the wavelength (p ≈ λ

_{o}/4) [18]. Therefore, for the axial mode, the pitch angle must be around 15 degrees, following Equation (1). This mode produces high directivity as illustrated in Figure 2c and provides circular polarization. The use of twist and glide symmetries, depicted in the following subsections, increase the propagation constant. This can be of interest for antenna miniaturization for a given frequency. The miniaturization of the helix antenna in axial mode is useful in applications with space restrictions, such as in antenna arrays.

_{p}) and gap between turns (g) regarding the basic helix parameters.

#### 2.2. Periodic Glide-Symmetrical and Twist-Symmetrical Unitary Cells

_{p}). Figure 3a shows the baseline cell both in 3D and in a planar view.

_{c}), its length (h

_{c}), and the number of corrugations per unit cell (N

_{c}). The unit cell is composed by sub-cells that consist of a pair of adjacent corrugations.

_{p}/N

_{subcell}. Therefore, the number of glide periods is N

_{subcell}= N

_{c}/2. It should be noted that, although this configuration is glide in its unwrapped version, once it is rolled, the relation between corrugations of the consecutive turns are not glide. This can be corrected by including a small misalignment (offset) of the corrugations in one of the strip sides so the corrugations of the structure, once wrapped, fit together again. The offset value that satisfies this condition is p·sin(α)-L

_{p}/N

_{c}as illustrated in Figure 3d. Notice that this case keeps a glide configuration, not regarding the strip, but between strip turns.

_{t}= 0) at these boundaries. This does not have influence in the frequency range under study since the distance to the box boundaries is much larger than the helix diameter. In the x direction (the direction of the axis of the helix structure), the boundary condition is periodic.

#### 2.3. Parametric Tuning Effects

#### 2.3.1. On the Twist Symmetry

_{c}) are depicted in Figure 4a. The increase in the length produces a higher propagation constant, which means an increase of the effective refractive index of the structure. Figure 4b provides the results of modifying the width w

_{c}, while preserving the number of corrugations per turn (N

_{c}). This effect is smaller than in the case of the length of the corrugations (h

_{c}). In Figure 4c, we illustrate the effect of the number of corrugations per turn (N

_{c}) for a fixed corrugation width. Although the cell periodicity is the same for all the cases, the different dimensions and number of corrugations introduce a variation in the propagation constant. The parametric study carried out in Figure 4c reveals that the increase in the number of corrugations per turn has a small influence in the propagation constant.

#### 2.3.2. On the Combined Twist and Glide Symmetry

_{c}). Variations in the width of the corrugations (Figure 5b) and the number of corrugations (Figure 5c) have a more limited effect.

#### 2.4. Symmetry Breakage

## 3. Helix Antenna Miniaturization

#### 3.1. Conventional Helix Antenna Design

_{turns}is chosen to be 12. A ground plane is located at the bottom of the antenna with a side dimension around one wavelength at the working frequency. A coaxial cable and a standard SMA-type transition, both of 50 Ω, are considered for the antenna feeding. These dimension values for the unit cell imply that the third mode is the one excited in our conventional antenna design.

_{11}| calculated in CST Microwave Studio. The circular polarization bandwidth is illustrated in grey. The circular polarization bandwidth represents the frequency range where the antenna has an axial ratio (AR) below 3 dB at the main direction of radiation and operates in axial mode.

_{11}|) of 54.5%, and an AR bandwidth (below 3dB) of 49%. Figure 7b–d illustrate the 3D radiation pattern of the helix antenna at different frequencies. An end-fire radiation pattern with a high directivity is achieved, as expected for the axial mode.

#### 3.2. Twist-And-Glide Symmetrical Helix Antenna Design

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Examples of higher symmetries: (

**a**) glide-symmetrical corrugations. The corrugations are periodic along x axis, and the mirroring plane is z = 0; (

**b**) Twist-symmetrical metallic rod with inclusions rotated φ = 90° along the twist axis.

**Figure 2.**Conventional helix antenna composed of a wire and a ground plane: (

**a**) antenna model and scheme; (

**b**) normal mode of radiation; (

**c**) axial mode of radiation.

**Figure 3.**Periodic helix cell: (

**a**) conventional helix (without corrugations); (

**b**) twist-symmetrical helix; (

**c**) combined twist- and glide-symmetrical helix; (

**d**) combined twist-and-glide-symmetric helix with offset in the glide configuration; (

**e**) Their dispersion diagrams. The reference dimensions are: r = 12 mm, α = 15°, p = 20.2 mm, L

_{p}= 78.06 mm, g = 12.78 mm, w = p/3, N

_{c}= 8, h

_{c}= 0.4g and w

_{c}= 0.3L

_{p}/N

_{c}.

**Figure 4.**Simulated dispersion diagrams for the twist-symmetrical helix cell: (

**a**) modification in the length of the corrugations (h

_{c}); (

**b**) modification of the width of the corrugations (w

_{c}), while preserving the number of corrugations per turn (N

_{c}= 4); (

**c**) modification of the number of corrugations (N

_{c}), for a given width, w

_{c}= 0.06L

_{p}. The reference dimensions are: r = 12 mm, α = 15°, p = 20.2 mm, L

_{p}= 78.06 mm, g = 12.78 mm, w = p/3, h

_{c}= 0.4g and w

_{c}= 0.3L

_{p}/N

_{c}.

**Figure 5.**Simulated dispersion diagrams for the twist- and glide-symmetrical helix cell: (

**a**) modification of the length of the corrugations (h

_{c}); (

**b**) modification of the width of the corrugations (w

_{c}), while preserving the number of glide periods (N

_{subcell}= 4) per turn; (

**c**) modification of the number of glide periods (N

_{subcell}), for a given width, w

_{c}= 0.06L

_{p}. The reference dimensions are: r = 12 mm, α = 15°, p = 20.2 mm, L

_{p}= 78.06 mm, g = 12.78 mm, w = p/3, h

_{c}= 0.4g and w

_{c}= 0.3L

_{p}/N

_{c}.

**Figure 6.**Symmetry breakage: (

**a**) twist-and-broken glide-symmetrical unit cell; (

**b**) dispersion diagram. The reference dimensions are: r = 12 mm, α = 15°, p = 20.2 mm, L

_{p}= 78.06 mm, g = 12.78 mm, w = p/3, N

_{c}= 8, h

_{c1}= 0.3g, h

_{c2}= 0.9g and w

_{c}= 0.3L

_{p}/N

_{c}.

**Figure 7.**Simulated results of a conventional helix antenna in axial mode: (

**a**) |S

_{11}| results; (

**b**) 3D radiation pattern in axial mode at 2 GHz (

**c**) 2.45 GHz and (

**d**) 2.8 GHz. The conventional helix unit cell dimensions are: r = 19.5 mm, α = 15°, p = 32.82 mm, w = p/3 and L

_{p}= 126.84 mm.

**Figure 8.**Simulated results of the miniaturized glide- and twist-symmetrical helix antenna centered at 2.45 GHz: (

**a**) Simulated |S

_{11}| (

**b**) 3D radiation pattern in axial mode at 2 GHz (

**c**) 2.45 GHz and (

**d**) 2.8 GHz. The twist-and-glide symmetrical unit cell dimensions are: r = 14.8 mm, α = 15°, p = 24.91 mm, w = p/3, L

_{p}= 96.27 mm, N

_{subcell}= 4, h

_{c}= 7.5 mm and w

_{c}= 0.3L

_{p}/N

_{subcell}.

**Figure 9.**(

**a**) Directivity comparison (

**b**) Antenna size comparison. The twist-and-glide symmetrical unit cell dimensions are: r = 14.8 mm, α = 15°, p = 24.91 mm, w = p/3, L

_{p}= 96.27 mm, N

_{subcell}= 4, h

_{c}= 7.5 mm and w

_{c}= 0.3L

_{p}/N

_{subcell}.

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**MDPI and ACS Style**

Palomares-Caballero, Á.; Padilla, P.; Alex-Amor, A.; Valenzuela-Valdés, J.; Quevedo-Teruel, O.
Twist and Glide Symmetries for Helix Antenna Design and Miniaturization. *Symmetry* **2019**, *11*, 349.
https://doi.org/10.3390/sym11030349

**AMA Style**

Palomares-Caballero Á, Padilla P, Alex-Amor A, Valenzuela-Valdés J, Quevedo-Teruel O.
Twist and Glide Symmetries for Helix Antenna Design and Miniaturization. *Symmetry*. 2019; 11(3):349.
https://doi.org/10.3390/sym11030349

**Chicago/Turabian Style**

Palomares-Caballero, Ángel, Pablo Padilla, Antonio Alex-Amor, Juan Valenzuela-Valdés, and Oscar Quevedo-Teruel.
2019. "Twist and Glide Symmetries for Helix Antenna Design and Miniaturization" *Symmetry* 11, no. 3: 349.
https://doi.org/10.3390/sym11030349