# Influence of Dielectric Plate Parameters on the Reflection Coefficient of a Planar Aperture Antenna

^{*}

## Abstract

**:**

_{11}of the antenna opening was systematically analyzed. The geometrical parameters taken in this analysis are the thickness d and the width/length h

_{1}/h

_{2}of the dielectric plate. The electromagnetic parameters used in this analysis are the real and the imaginary part of permittivity (ε

_{r}, tan δ) and the electrical conductivity of the dielectric plate (σ). The simulation calculation and analysis included other structural and electromagnetic parameters of the dielectric plate (density of the radome material, relative permeability, and magnetic loss tangent (ρ, µ

_{r}, and tan δ

_{µ}, respectively)), but the results show that in the range of real values of these parameters for the materials used for the dielectric plate, they had no significant influence on the reflection coefficient. The results show that impedance-matched antennas with very low values of the reflection coefficients S

_{11}at the resonant frequency can be realized by changing the geometrical and electromagnetic parameters of the dielectric plate material. The results are presented for a circular aperture antenna on a planar grounded plane with a dielectric plate on the opening, and the achieved lowest values of the S

_{11}parameter were −45.17 dB (simulated) and −43.93 dB (measured) at the frequencies of 1.7820 GHz and 1.7550 GHz, respectively. The estimated values of the dielectric plate parameters in this case are thickness d = 11.08 mm (0.67 λ); width × length of grounded plane and dielectric plate h

_{1}× h

_{2}= 423 × 450 mm

^{2}(2.51 × 2.67 λ); relative permittivity 2.5, tan δ = 0.09, μ

_{r}= 1, tan δ

_{μ}= 0.00, ρ = 1200 kg·m

^{−3}; and electrical conductivity σ = 0 S/m. The simulation calculation results were verified by measuring the reflection coefficient S

_{11}on the created laboratory model of the aperture antenna with the dielectric plate and showed a very good match.

## 1. Introduction

_{11}of an aperture antenna fed by a circular waveguide covered with a dielectric plate was Ansys HFSS (three-dimensional electromagnetic (EM) simulation software for the design and simulation of HF electronic element–antennas and antenna arrays). The theoretical and mathematical basis of the aforementioned software for solving problems in electromagnetism is the finite element method.

^{−3}, relative permittivity ε

_{r}, conductivity of the dielectric σ in S/m, losses in the dielectric tan δ and tan δ

_{μ}, and relative permeability of the material µ

_{r}), and finally, simulation is validated with measured values. The aim of the article is to achieve the lowest possible reflection coefficient of the circular aperture antenna in order to accomplish the best impedance matching of the antenna itself.

_{11}of the TE

_{11}mode aperture antenna fed by a circular waveguide covered with dielectric is simulated and measured.

## 2. Analysis of the Parameters of the Dielectric Plate on the Antenna Aperture

#### 2.1. The Thickness and Width × Length of the Antenna Aperture Dielectric Plate

#### 2.1.1. Propagation and Reflection of a Plane Wave at the Boundary of Two Mediums

_{i}and H

_{i}), whereas the wave that passes the boundary under certain conditions is called the transmitted wave (E

_{t}and H

_{t}). At the same time, part of the wave is reflected at the boundary and is called a reflected wave (E

_{r}and H

_{r}). A simplified situation is shown in Figure 1.

_{11}is calculated according to the following relation:

_{r}—the electric field strength of the reflected wave in V/m;

_{i}—the electric field strength of the incident wave in V/m;

_{c1}= Z

_{c3}—characteristic impedance of the first and the third medium—the impedance of air in the waveguide is equal to the impedance of free space Z

_{0}in Ω;

_{c2}—characteristic impedance of the second medium–dielectric plate, also in (Figure 1).

_{i}—the i-th medium permeability;

_{i}—the i-th medium permittivity.

_{1}= ε

_{0}(the permittivity of the air in the waveguide), and ε

_{2}is the dielectric constant of the dielectric plate; (b) for the second boundary, conversely, ε

_{1}is the permittivity of the dielectric plate, and ε

_{2}= ε

_{0}. In the geometry of the two boundaries (the space inside the waveguide–dielectric plate and the dielectric plate-free space), the electric field strength E (incident, transmitted, and reflected) in each of the mentioned mediums is presented in detail in Figure 2a.

_{i}passes the boundary between two mediums (E

_{t}), whereas part of the wave is reflected (E

_{r}).

_{11}for two mediums is (Figure 2b) as follows:

_{1}—the first medium reflection coefficient;

_{11}—the total reflection coefficient;

_{1}—the wave number of the first medium.

#### 2.1.2. Propagation and Reflection of an EM Wave Generated in a Circular Cross-Section Waveguide at the Boundary of Two Mediums

_{11}. Figure 4 shows the E and H field lines of the dominant mode of a circular cross-section waveguide The straight line of the arrow indicates the strength of the electric field E. The dashed arrow indicates the strength of the magnetic field H (the H-field is perpendicular to the E-field). The signs × and dot indicate the direction of the H-field—inwards and outwards respectively.

_{11}mode electric field inside the waveguide can be expressed by the equation:

_{11}mode electric field at the waveguide aperture (antenna aperture) [15]:

_{i}= 0°. It can be easily concluded that the other two angles are Θ

_{r}= 0°and Θ

_{t}= 0°.

_{11}parameter, whereas the width/length of the dielectric plate is about the connection between the majority of the grounded plane and possible diffraction problems.

_{11}with different parameters of the protective shield (radome).

#### 2.2. Permittivity and Electrical Conductivity of the Dielectric Plate

^{−jδ}, where ε′ = ε

_{0}ε

_{r}is the real part of the permittivity ε that represents the stored electric field, and ε″= ε

_{0}ε

_{r}″ is the imaginary part of the permittivity that stands for the loss factor-a measure of the amount of energy dissipated. δ is the dielectric loss angle. The parameter tan δ is loss tangent or the dissipation factor DF.

_{11}depends on the complex electrical permittivity, the complex magnetic permeability, and layer thickness. Because the moisture of the material has an effect on the imaginary part of the complex permittivity, in addition to the previously mentioned parameters, S

_{11}is also affected by moisture (electrical conductivity) and temperature [15]:

_{11}is the reflection coefficient, $\mathsf{\epsilon}\prime $ and $\mathsf{\epsilon}\u2033$ are the real and the imaginary part of the complex electric permittivity, $\mathsf{\mu}\prime $ and $\mathsf{\mu}\u2033$ are the real and the imaginary part of the complex magnetic permeability, f is the frequency of an electromagnetic wave, c is the speed of light, and d is material thickness.

## 3. Numerical Results

_{11}. The parameters are divided into structural–geometric (thickness d and dielectric density ρ) and electromagnetic parameters (relative dielectric constant ε

_{r}, dielectric losses tg δ and tg δ

_{μ}, dielectric conductivity σ, and relative dielectric permeability µ

_{r}).

_{1}, length h

_{2}) of the grounded plane and dielectric plate are h

_{1}= 423 mm (2.51 λ) and h

_{2}= 450 mm (2.67 λ), and the radius and length of the waveguide are r

_{w}= 61 mm (0.276 λ) and l

_{w}= 171 mm (1.02 λ), respectively. The space inside the waveguide is defined by ε

_{1}and µ

_{1}(the waveguide is air-filled). The dielectric plate is characterized by ε

_{2}, µ

_{2}, and the thickness d. The space following behind the dielectric plate is free space, and it is characterized by ε

_{0}, µ

_{0}.

_{11}with the corresponding cut-off frequency f

_{cTE11}. The next mode is TM

_{01}with the cut-off frequency f

_{cTM01}. Therefore, numerical calculations were carried out in the frequency range f

_{cTE11}< f < f

_{cTM01}.

#### 3.1. The Influence of the Structural–Geometric Parameters of the Dielectric on S_{11}

_{1}× h

_{2}, and the density ρ of the dielectric plate in kg·m

^{−3}.

#### 3.1.1. Influence of Dielectric Plate Thickness d on the Reflection Coefficient

_{11}at the opening of the waveguide. The thickness d ranges from d = 0 mm to d = 13 mm, assuming all other parameters: ε

_{r}= 2.5, µ

_{r}= 1, σ = 0 S/m, tg δ = 0.09, tg δ

_{μ}= 0. Some values of these parameters were carefully selected according to [11,17,18].

_{11}decreases with increasing dielectric plate thickness d, namely from −12.57 dB (d = 0 mm) to −43.43 dB (d = 11 mm). After that, for dielectric plate thickness of 13 mm, the value of the reflection coefficient increases (−27.10 dB). This is consistent with the theoretical considerations in Section 2 and the minimum absolute value of the parameter S

_{11}(between 1 and 2 GHz) shown in Figure 2b. It can be seen that the resonant frequency increases (shifts to the right) with increasing dielectric plate thickness at the antenna aperture. After reaching dielectric plate thickness of about 11 mm (and the minimum value of parameter S

_{11}), the resonant frequency decreases with increasing dielectric plate thickness. The reason for this is that the reflected waves from the first (air-dielectric plate) and the second (dielectric plate-air) boundary at this thickness (about 11 mm) are in destructive interference, resulting in the minimum value of the reflection coefficient S

_{11}. Figure 7 and Table 1 show the change in the reflection coefficient as a function of the change in the thickness of the dielectric d. The minimum values of the coefficient S

_{11}for certain values of d are listed in Table 1 together with the associated frequency f.

#### 3.1.2. Influence of the Dielectric Plate Dimension h_{1} × h_{2} (Width × Length) on the Reflection Coefficient and the Radiation Pattern of the Antenna Opening

_{11}were performed to change the dimensions of the dielectric plate h

_{1}× h

_{2}by ±20%, which resulted in negligible changes in the reflection coefficient S

_{11}.

#### 3.1.3. Influence of Dielectric Plate Density ρ on the Reflection Coefficient

^{−3}. The calculations showed a constant value of the reflection coefficient S

_{11}= −43.43 dB (f = 1.7820 GHz) for ρ $\in $ [500, 1800] kg·m

^{−3}. Therefore, for all further simulations of the reflection coefficient according to [19], it is assumed that ρ = 1200 kg·m

^{−3}.

#### 3.2. The Influence of the Electromagnetic Parameters of the Dielectric on S_{11}

#### 3.2.1. Influence of the Dielectric Plate Relative Permittivity ε_{r} on the Reflection Coefficient

_{11}for ε

_{r}$\in $ [1.5, 10] were carried out under the condition that the thickness of the dielectric d is fixed at 11 mm (the best value from the previous subsection). The other dielectric parameters remain as given in Section 3.1.1. The simulation results are presented in Figure 8 and Table 2.

_{11}is minimal at the dielectric constant ε

_{r}= 2.5, which, according to the literature cited above, proves that a material with such dielectric constant value is an excellent choice for the antenna radome.

_{11}(f) and ε

_{r}= const, which also corresponds to the resonant frequency of the aperture antenna, increases (it moves to the right towards higher frequencies) with an increase in the permittivity constant. This increase in resonance occurs up to the value of the permittivity 2.5, at which the value of the reflection coefficient S

_{11}reaches the minimum value of −43.43 dB at the frequency 1.7820 GHz. With a further increase in the permittivity, the value of the resonant frequency decreases, whereas the minimum value of the curve S

_{11}(f), ε

_{r}= const, increases.

#### 3.2.2. Influence of Dielectric Plate Electric Conductivity σ on the Reflection Coefficient

_{11}with a change in conductivity σ $\in $ [0, 10]. The calculated values of S

_{11}are given in Table 3 and shown in Figure 9. The other dielectric parameters remain as given in Section 3.1.1, with d = 11 mm and ε

_{r}= 2.5 (the last best achieved values).

_{11}= −43.43 dB at a dielectric conductivity of 0 S/m. With increasing conductivity values, S

_{11}decreases drastically (for example, for σ = 1 S/m, S

_{11}= −4.85 dB).

^{−1}to 10 S/m), the electrical conductivity component prevails, and the material is no longer considered a good dielectric. Therefore, the reflection coefficient increases significantly, as expected for a conductive material. Thus, due to moisture in the dielectric material, which causes a change in electrical conductivity of the material, and depending on moisture content, electrical conductivity can cause a significant change in the reflection coefficient S

_{11}.

#### 3.2.3. Influence of Dielectric Plate Dielectric Losses tan δ on the Reflection Coefficient

_{11}were carried out, and the best value was achieved for tan δ = 0.09—Figure 10 (i.e., S

_{11}= −43.43 dB).

#### 3.2.4. Influence of Dielectric Plate Relative Permeability μ_{r} and Magnetic Loss Tangent tan δ_{µ} on the Reflection Coefficient

_{0}1 − j 0 (relative permeability is equal to 1, and plate magnetic loss tangent tan δ

_{µ}= 0).

_{r}ranging from 0.9 to 1.1, and tan δ

_{µ}ranging from 0 to 2), and they showed that a deviation from the defined values μ

_{r}= 1 and tan δ

_{µ}= 0 significantly degrades the value of the reflection coefficient S

_{11}.

#### 3.3. Simulation Calculation of the Radiation Pattern of a Circular Aperture Antenna with a Dielectric Plate

_{r}= 2.5, µ

_{r}= 1, σ = 0 S/m, tg δ = 0.09, tg δ

_{μ}= 0, h

_{1}× h

_{2}= 423 × 450 mm

^{2}, ρ = 1200 kg·m

^{−3}), it can be concluded that there are some changes in the radiation diagram. The largest deviations of −2.5 dB at ±25° (side lobes) and +5.0 dB at 158° (back lobe) can be seen in the H plane and +9.0 dB at 165° (back lobes) in the E plane, whereas deviations are negligible in the rest of the range of angles in both planes (E and H planes).

_{1}× h

_{2}= 423 × 450 mm

^{2}), does not result in significant changes in the radiation pattern compared to the dielectric plate with dimension h

_{1}× h

_{2}= 423 × 450 mm

^{2}(Figure 11). The largest deviations of +4.5 dB at ±158° (back lobes) can be seen in the H plane, and +6.0 dB at 117 and 168° (back lobe) in the E plane, whereas deviations are negligible in the rest of the range of angles in both planes (E and H planes).

^{2}(2.51 × 2.67 λ) has an effect on the side and back lobes, but there is no main lobe and, therefore, it does not significantly affect the operation of this antenna. Lowering the side lobe in the H plane further slightly improves the radiation pattern.

## 4. Measurement Results

_{11}, must be validated via laboratory measurements. What can be measured with certainty is S

_{11}for a given thickness d of the dielectric plate on the antenna. Laboratory measurements of S

_{11}at the waveguide–free space boundary and the waveguide–dielectric boundary were performed for d = 0 mm, d = 4.78 mm, d = 7.08 mm, d = 9.26 mm, and d = 11.08 mm. Simulation calculation precedes the measurements for these dielectric plate thickness values by using paper as a dielectric. Comparison results of the simulated and measured values of S

_{11}are shown in Figure 12a–e. Table 6 shows the comparative values of the reflection coefficients (simulated and measured) for different values of dielectric thickness.

_{11}at the boundary between a waveguide-dielectric and a dielectric-free space was carried out by using an N9914A FieldFox Handheld RF spectrum analyzer, the operating range of which extends up to 6.5 GHz. The laboratory in which the reflection coefficient measurement was performed is not a chamber without an electromagnetic echo (Figure 13).

## 5. Discussion

_{11}can be significantly affected, and with the proper selection of all dielectric parameters, this coefficient value can be brought to the minimum value. Figure 14 clearly shows the situations in which no dielectric plate is placed on the antenna and in which a dielectric plate with the best parameter values achieved (d = 11 mm, ε

_{r}= 2.5, µ

_{r}= 1, ρ = 1200 kg·m

^{−3}, σ = 0 S/m, tan δ = 0.09, tan δ

_{μ}= 0) is placed on the antenna.

_{1}/h

_{2}of the dielectric plate. The electromagnetic parameters used in this analysis are the real and the imaginary part of permittivity (ε

_{r}, tan δ) and electrical conductivity of the dielectric plate (σ). Simulation calculation and analysis included other structural and electromagnetic parameters of the dielectric plate (density of the dielectric material, relative permeability, and magnetic loss tangent (ρ, µ

_{r}, and tan δ

_{µ})), but the results showed that in the range of real values of dielectric parameters for the materials used for the dielectric plate, those three parameters had no significant influence on the reflection coefficient. Finally, simulation is validated with measured values. The aim of the study was to achieve the lowest possible reflection coefficient of the circular aperture antenna in order to accomplish the best impedance matching of the antenna itself.

_{11}= −43.43 dB is obtained for d = 11 mm.

_{r}ranges from 1 to 10, and the best result of S

_{11}is obtained for ε

_{r}= 2.5. This ε

_{r}value corresponds to the relative permittivity of the paper used as a dielectric material in the measurements, but such permittivity has some other materials suitable to be used for the antenna radome.

_{11}. This statement is also evident in Figure 12a–e, in which the graphs of the measured values show a frequency shift to the left, i.e., to lower frequency values. Another circumstance favors the difference between the simulated and the measured values; in practice, it is very difficult to adjust dielectric parameters to the value used by the simulation. The only parameter value that can be measured very accurately is the dielectric thickness d.

_{11}of −43.93 dB at the frequency of 1.7550 GHz. The minimum reflection coefficient obtained by additional simulation for d = 11.08 mm is −45.17 dB at the frequency of 1.7820 GHz. Apart from d, which is variable, other parameters are set within the limits given in [17,20].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Propagation of an EM wave through a dielectric plate of thickness d (one dielectric layer): (

**a**) existing electric field strengths (incident, transmitted, and reflected); (

**b**) the absolute value of the reflection and transmission coefficient S

_{11}.

**Figure 5.**Electromagnetic wave propagates through three mediums: space in the wave-guide, a dielectric plate on the aperture antenna, and free space.

**Figure 6.**(

**a**) Block diagram of the geometry of the problem; (

**b**) HFSS model of an aperture antenna fed by a circular waveguide placed on a grounded planar surface with dielectric plate.

**Figure 7.**The simulated reflection coefficient S

_{11}for an aperture antenna with plate width in the range of d = 1–13 mm.

**Figure 8.**The simulated reflection coefficient S

_{11}for an aperture antenna with dielectric plate permittivity ε

_{r}$\in $ [1.5, 3.1].

**Figure 9.**The simulated reflection coefficient S

_{11}for an aperture antenna with dielectric plate conductivity σ $\in $ [0, 10].

**Figure 10.**The simulated reflection coefficient S

_{11}for an aperture antenna with dielectric plate tan δ $\in $ [0.01, 0.09].

**Figure 11.**Radiation pattern of an aperture antenna with a dielectric plate: (

**a**) E plane; (

**b**) H plane.

**Figure 12.**Simulated and measured reflection coefficient S

_{11}for an aperture antenna: (

**a**) without dielectric plate (d = 0 mm) and with dielectric plate thickness: (

**b**) d = 4.78 mm, (

**c**) d = 7.08 mm, (

**d**) d = 9.26 mm, and (

**e**) d = 11.08 mm.

**Figure 13.**(

**a**) Measurement of the S

_{11}coefficient of the aperture antenna fed by a circular waveguide (covered with dielectric material) with an N9914A FieldFox Handheld RF Analyzer (6.5 GHz); (

**b**) side view; (

**c**) aperture antenna without radome; (

**d**) space inside the waveguide.

**Figure 14.**Simulated reflection coefficient S

_{11}for an aperture antenna with a dielectric plate of thickness d = 11 mm and without dielectric plate (d = 0 mm).

d, mm | f, GHz | S_{11}, dB |
---|---|---|

0 | 1.7011 | −12.57 |

1 | 1.7213 | −12.60 |

3 | 1.7550 | −13.19 |

5 | 1.7753 | −15.21 |

7 | 1.7820 | −19.81 |

9 | 1.7820 | −25.68 |

11 (0.67 λ) | 1.7820 | −43.43 |

13 | 1.7753 | −27.10 |

ε_{r} | f, GHz | S_{11}, dB |
---|---|---|

1.5 | 1.7483 | −14.90 |

1.9 | 1.7685 | −18.55 |

2.1 | 1.7753 | −21.74 |

2.5 | 1.7820 | −43.43 |

2.9 | 1.7820 | −23.01 |

3.1 | 1.7820 | −19.38 |

5 | 1.7753 | −08.22 |

7 | 1.7685 | −05.03 |

10.0 | 1.7550 | −03.24 |

σ, S/m | f, GHz | S_{11}, dB |
---|---|---|

0 | 1.7820 | −43.43 |

$1.0\times {10}^{-12}$ | 1.7820 | −43.43 |

$1.0\times {10}^{-3}$ | 1.7820 | −43.41 |

$1.0\times {10}^{-1}$ | 1.7550 | −21.55 |

1 | 1.7348 | −4.85 |

10 | 1.7416 | −1.59 |

**Table 4.**Verification of condition: $\frac{\sigma}{\omega {\epsilon}_{0}}$ << ${\mathsf{\epsilon}}_{\mathrm{d}}^{\u2033}$ of simulation results for different dielectric plate conductivity σ, according to Equation (11).

σ, S/m | $\frac{\mathsf{\sigma}}{\mathsf{\omega}{\mathsf{\epsilon}}_{0}}$; (f = 1.0 GHz) | $\frac{\mathsf{\sigma}}{\mathsf{\omega}{\mathsf{\epsilon}}_{0}}$; (f = 2.5 GHz) | ${\mathsf{\epsilon}}_{\mathbf{d}}^{\u2033}$ = 2.5 tanδ | S_{11}, dB |
---|---|---|---|---|

0 | 0 | 0 | 0.225 | −43.43 |

$1.0\times {10}^{-12}$ | $1.80\times {10}^{-11}$ | $7.19\times {10}^{-12}$ | 0.225 | −43.43 |

$1.0\times {10}^{-3}$ | $1.8\times {10}^{-2}$ | $7.19\times {10}^{-3}$ | 0.225 | −43.41 |

$1.0\times {10}^{-1}$ | $1.8$ | $7.19\times {10}^{-1}$ | 0.225 | −21.55 |

1 | $1.8\times {10}^{+1}$ | $7.19$ | 0.225 | −4.85 |

10 | $1.8\times {10}^{+2}$ | $7.19\times {10}^{+1}$ | 0.225 | −1.59 |

tg δ | f, GHz | S_{11}, dB |
---|---|---|

0.010 | 1.7887 | −41.48 |

0.030 | 1.7887 | −41.48 |

0.050 | 1.7887 | −34.41 |

0.060 | 1.7820 | −35.19 |

0.070 | 1.7820 | −39.01 |

0.090 | 1.7820 | −43.43 |

0.098 | 1.7820 | −41.48 |

**Table 6.**Comparative values of the reflection coefficient S

_{11}for an antenna with and without dielectric plate at the antenna aperture.

d, mm | f_{sim}, GHz | S_{11sim}, dB | f_{meas}, GHz | S_{11meas}, dB |
---|---|---|---|---|

0 | 1.6944 | −12.57 | 1.7147 | −13.15 |

4.78 | 1.7753 | −15.23 | 1.7550 | −16.81 |

7.08 | 1.7820 | −19.99 | 1.7550 | −24.41 |

9.26 | 1.7820 | −27.73 | 1.7600 | −25.49 |

11.08 | 1.7820 | −45.17 | 1.7550 | −43.93 |

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**MDPI and ACS Style**

Mandrić, V.; Rupčić, S.; Rimac-Drlje, S.; Baxhaku, I. Influence of Dielectric Plate Parameters on the Reflection Coefficient of a Planar Aperture Antenna. *Appl. Sci.* **2023**, *13*, 2544.
https://doi.org/10.3390/app13042544

**AMA Style**

Mandrić V, Rupčić S, Rimac-Drlje S, Baxhaku I. Influence of Dielectric Plate Parameters on the Reflection Coefficient of a Planar Aperture Antenna. *Applied Sciences*. 2023; 13(4):2544.
https://doi.org/10.3390/app13042544

**Chicago/Turabian Style**

Mandrić, Vanja, Slavko Rupčić, Snježana Rimac-Drlje, and Ismail Baxhaku. 2023. "Influence of Dielectric Plate Parameters on the Reflection Coefficient of a Planar Aperture Antenna" *Applied Sciences* 13, no. 4: 2544.
https://doi.org/10.3390/app13042544