# Theoretical Study of the Input Impedance and Electromagnetic Field Distribution of a Dipole Antenna Printed on an Electrical/Magnetic Uniaxial Anisotropic Substrate

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Formulation

## 3. Method of Solution

**1st region:**$${\tilde{E}}_{x1}\left(\alpha ,\beta ,z\right)=\frac{j}{{\alpha}^{2}+{\beta}^{2}}\frac{1}{\omega {\epsilon}_{0}}\left(-\alpha {\gamma}_{e}^{2}{\gamma}_{0}\mathrm{Se}\times {A}_{e}+\beta {\kappa}_{0}^{2}{\mu}_{t}\mathrm{Sh}\times {A}_{h}\right)$$$${\tilde{E}}_{y1}\left(\alpha ,\beta ,z\right)=\frac{j}{{\alpha}^{2}+{\beta}^{2}}\frac{1}{\omega {\epsilon}_{0}}\left(-\beta {\gamma}_{e}^{2}{\gamma}_{0}\mathrm{Se}\times {A}_{e}-\alpha {\kappa}_{0}^{2}{\mu}_{t}\mathrm{Sh}\times {A}_{h}\right)$$$${\tilde{E}}_{z1}\left(\alpha ,\beta ,z\right)=-\frac{{\gamma}_{0}{\gamma}_{ec}^{}{\epsilon}_{t}}{\omega {\epsilon}_{0}{\epsilon}_{z}}\mathrm{Se}\times {A}_{e}$$$${\tilde{H}}_{x1}\left(\alpha ,\beta ,z\right)=\frac{1}{{\alpha}^{2}+{\beta}^{2}}\left(\beta {\gamma}_{0}{\epsilon}_{t}{\gamma}_{ec}^{}\mathrm{Se}\times {A}_{e}-\alpha {\gamma}_{hc}^{}\mathrm{Sh}\times {A}_{h}\right)$$$${\tilde{H}}_{y1}\left(\alpha ,\beta ,z\right)=\frac{1}{{\alpha}^{2}+{\beta}^{2}}\left(-\alpha {\gamma}_{0}{\epsilon}_{t}{\gamma}_{ec}^{}\mathrm{Se}\times {A}_{e}-\beta {\gamma}_{hc}^{}\mathrm{Sh}\times {A}_{h}\right)$$$${\tilde{H}}_{z1}\left(\alpha ,\beta ,z\right)=j\frac{{\mu}_{t}}{{\mu}_{z}}\mathrm{Sh}\times {A}_{h}$$**2nd region:**$${\tilde{E}}_{x2}\left(\alpha ,\beta ,z\right)=j\frac{{e}^{-{\gamma}_{0}\left(z-d\right)}}{{\alpha}^{2}+{\beta}^{2}}\frac{1}{\omega {\epsilon}_{0}}\left[-\alpha {\gamma}_{0}{\gamma}_{e}^{2}{A}_{e}+\beta {\mu}_{t}{\kappa}_{0}^{2}{A}_{h}\right]$$$${\tilde{E}}_{y2}\left(\alpha ,\beta ,z\right)=j\frac{{e}^{-{\gamma}_{0}\left(z-d\right)}}{{\alpha}^{2}+{\beta}^{2}}\frac{1}{\omega {\epsilon}_{0}}\left[-\beta {\gamma}_{0}{\gamma}_{e}^{2}{A}_{e}-\alpha {\mu}_{t}{\kappa}_{0}^{2}{A}_{h}\right]$$$${\tilde{E}}_{z2}\left(\alpha ,\beta ,z\right)=\frac{{\gamma}_{e}^{2}}{\omega {\epsilon}_{0}}{A}_{e}{e}^{-{\gamma}_{0}\left(z-d\right)}$$$${\tilde{H}}_{x2}\left(\alpha ,\beta ,z\right)=\frac{{e}^{-{\gamma}_{0}\left(z-d\right)}}{{\alpha}^{2}+{\beta}^{2}}\left(-\beta {\gamma}_{e}^{2}{A}_{e}+{\mu}_{t}\alpha {\gamma}_{0}{A}_{h}\right)$$$${\tilde{H}}_{y2}\left(\alpha ,\beta ,z\right)=\frac{{e}^{-{\gamma}_{0}\left(z-d\right)}}{{\alpha}^{2}+{\beta}^{2}}\left(\alpha {\gamma}_{e}^{2}{A}_{e}+\beta {\mu}_{t}{\gamma}_{0}{A}_{h}\right)$$$${\tilde{H}}_{z2}\left(\alpha ,\beta ,z\right)=j{\mu}_{t}{A}_{h}{e}^{-{\gamma}_{0}\left(z-d\right)}$$$${A}_{e}=\frac{\alpha {\tilde{J}}_{x}+\beta {\tilde{J}}_{y}}{\left({\gamma}_{e}^{2}+{\gamma}_{0}{\epsilon}_{t}{\gamma}_{e}\mathrm{coth}\left({\gamma}_{e}d\right)\right)}$$$${A}_{h}=\frac{\beta {\tilde{J}}_{x}-\alpha {\tilde{J}}_{y}}{\left({\gamma}_{h}\mathrm{coth}\left({\gamma}_{h}d\right)+{\gamma}_{0}{\mu}_{t}\right)}$$$${\gamma}_{ec}^{}={\gamma}_{e}\mathrm{coth}\left({\gamma}_{e}d\right)$$$${\gamma}_{hc}^{}={\gamma}_{h}\mathrm{coth}\left({\gamma}_{h}d\right)$$$$\mathrm{Se}=\frac{\mathrm{sinh}\left({\gamma}_{e}z\right)}{\mathrm{sinh}\left({\gamma}_{e}d\right)}$$$$\mathrm{Sh}=\frac{\mathrm{sinh}\left({\gamma}_{h}z\right)}{\mathrm{sinh}\left({\gamma}_{h}d\right)}$$

## 4. Fields Computations

^{®}software [44] is used to plot the fields distributions.

## 5. Numerical Results

#### 5.1. Validation

_{0}thick substrate planar dipole antenna. The objective of this work is to analyze the effects of different electromagnetic parameters of the anisotropic substrate on the input impedance of the dipole, in addition to the electromagnetic field evaluation through the plotting of the electric and magnetic field distributions in the three principal planes XY, XZ, and YZ.

_{0}as a function of normalized length L/λ

_{0}. These results represent a validation step of the accuracy of our calculations for both isotropic and anisotropic substrates. The representation shows good agreement with the data reported in [29]. In [29], only cases of electrical anisotropy were considered and no discussion of the effect of this component was conducted.

#### 5.2. Electromagnetic-Field Distributions in Isotropic Case

_{t}and H

_{t}, in the transverse plane with respect to z-, y- and x-axis, respectively, for the isotropic case. The arrow indicates the cross-sectional field vector direction and the arrow length designates the field magnitude, and the lines indicates the equi-phase field contour forms.

#### 5.3. Effect of the Electrical Uniaxial Anisotropy on the Electromagnetic-Field Distributions

#### 5.4. Effect of the Magnetic Uniaxial Anisotropy on Electromagnetic-Field Distributions

#### 5.5. Effect of the Uniaxial Anisotropy on Input Impedance

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**(

**a**–

**f**): Fields distribution plots in the transverse plane for the isotropic case (${\epsilon}_{t}$ = ${\epsilon}_{z}$ = 3.25 and ${\mu}_{t}$ = ${\mu}_{z}$ = 1).

**Figure 4.**(

**a**–

**f**) Normalized electric field distributions for various values of ${\epsilon}_{z}$ with ${\epsilon}_{t}=3.25$, ${\mu}_{z}=1$ and ${\mu}_{t}=1$ in the XY, XZ, and YZ planes.

**Figure 5.**(

**a**–

**f**) Normalized electric field distributions for various values of ${\epsilon}_{t}$ with ${\epsilon}_{z}=3.25$, ${\mu}_{z}=1$ and ${\mu}_{t}=1$ in the XY, XZ, and YZ planes.

**Figure 6.**(

**a**–

**f**) Normalized magnetic field distributions for various values of ${\epsilon}_{z}$ with ${\epsilon}_{t}=3.25$, ${\mu}_{z}=1$ and ${\mu}_{t}=1$ in the XY, XZ and YZ planes.

**Figure 7.**(

**a**–

**f**) Normalized magnetic field distributions for various values of ${\epsilon}_{t}$ with ${\epsilon}_{z}=3.25$, ${\mu}_{z}=1$ and ${\mu}_{t}=1$ in the XY, XZ, and YZ planes.

**Figure 8.**(

**a**–

**f**) Normalized electric field distributions for various values of ${\mu}_{z}$ with ${\epsilon}_{z}={\epsilon}_{t}=3.25$ and ${\mu}_{t}=1$, in the XY, XZ, and YZ planes.

**Figure 9.**(

**a**–

**f**) Normalized electric field distributions for various values of ${\mu}_{t}$ with ${\epsilon}_{z}={\epsilon}_{t}=3.25$ and ${\mu}_{z}=1$, in the XY, XZ, and YZ planes.

**Figure 10.**(

**a**–

**f**) Normalized magnetic field distributions for various values of ${\mu}_{z}$ with ${\epsilon}_{z}={\epsilon}_{t}=3.25$ and ${\mu}_{t}=1$, in the XY, XZ, and YZ plane.

**Figure 11.**(

**a**–

**f**) Normalized magnetic field distributions for various values of ${\mu}_{t}$ with ${\epsilon}_{z}={\epsilon}_{t}=3.25$ and ${\mu}_{z}=1$, in the XY, XZ, and YZ plane.

**Figure 12.**Real and imaginary parts of the input impedance for various values of (

**a**): ${\epsilon}_{z}$ and (

**b**): ${\epsilon}_{t}$.

**Figure 13.**Real and imaginary parts of the input impedance for various values of (

**a**): ${\mu}_{\mathrm{z}}$ and (

**b**): ${\mu}_{\mathrm{z}}$.

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**MDPI and ACS Style**

Bouknia, M.L.; Zebiri, C.; Sayad, D.; Elfergani, I.; Rodriguez, J.; Alibakhshikenari, M.; Abd-Alhameed, R.A.; Falcone, F.; Limiti, E.
Theoretical Study of the Input Impedance and Electromagnetic Field Distribution of a Dipole Antenna Printed on an Electrical/Magnetic Uniaxial Anisotropic Substrate. *Electronics* **2021**, *10*, 1050.
https://doi.org/10.3390/electronics10091050

**AMA Style**

Bouknia ML, Zebiri C, Sayad D, Elfergani I, Rodriguez J, Alibakhshikenari M, Abd-Alhameed RA, Falcone F, Limiti E.
Theoretical Study of the Input Impedance and Electromagnetic Field Distribution of a Dipole Antenna Printed on an Electrical/Magnetic Uniaxial Anisotropic Substrate. *Electronics*. 2021; 10(9):1050.
https://doi.org/10.3390/electronics10091050

**Chicago/Turabian Style**

Bouknia, Mohamed Lamine, Chemseddine Zebiri, Djamel Sayad, Issa Elfergani, Jonathan Rodriguez, Mohammad Alibakhshikenari, Raed A. Abd-Alhameed, Francisco Falcone, and Ernesto Limiti.
2021. "Theoretical Study of the Input Impedance and Electromagnetic Field Distribution of a Dipole Antenna Printed on an Electrical/Magnetic Uniaxial Anisotropic Substrate" *Electronics* 10, no. 9: 1050.
https://doi.org/10.3390/electronics10091050