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

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## 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

## References

- Kirilenko, A.A.; Steshenko, S.O.; Derkach, V.N.; Prikolotin, S.A.; Kulik, D.Y.; Prosvirnin, S.; Mospan, L.P. Rotation of the polarization plane by double-layer planar-chiral structures. Review of the results of theoretical and experimental studies. Radioelectron. Commun. Syst.
**2017**, 60, 193–205. [Google Scholar] [CrossRef] - Akdagli, A. Behaviour of Electromagnetic Waves in Different Media and Structures, 1st ed.; BoD–Books on Demand: Rijeka, Croatia, 2011. [Google Scholar]
- Guo, B. Photonic band gap structures of obliquely incident electromagnetic wave propagation in a one-dimension absorptive plasma photonic crystal. Phys. Plasmas
**2009**, 16, 043508. [Google Scholar] [CrossRef] - Krowne, C.M. Left-handed material anisotropy effect on guided wave electromagnetic fields. J. Appl. Phys.
**2006**, 99, 044914. [Google Scholar] [CrossRef] - Krowne, C.M.; Daniel, M. Electromagnetic field behavior in dispersive isotropic negative phase velocity/negative refractive index guided wave structures compatible with millimeter-wave monolithic integrated circuits. J. Nanomater.
**2007**, 2007, 054568. [Google Scholar] [CrossRef] - Krowne, C.M. Electromagnetic-field theory and numerically generated results for propagation in left-handed guided-wave single-microstrip structures. IEEE Trans. Microw. Theory Tech.
**2003**, 51, 2269–2283. [Google Scholar] [CrossRef] - Krowne, C.M. Electromagnetic distributions demonstrating asymmetry using a spectral-domain dyadic Green’s function for ferrite microstrip guided-wave structures. IEEE Trans. Microw. Theory Tech.
**2005**, 53, 1345–1361. [Google Scholar] [CrossRef] - Zebiri, C.; Lashab, M.; Benabdelaziz, F. Effect of anisotropic magneto-chirality on the characteristics of a microstrip resonator. IET Microw. Antennas Propag.
**2010**, 4, 446–452. [Google Scholar] [CrossRef] - Zebiri, C.; Lashab, M.; Benabdelaziz, F. Rectangular microstrip antenna with uniaxial bi-anisotropic chiral substrate–superstrate. IET Microw. Antennas Propag.
**2011**, 5, 17–29. [Google Scholar] [CrossRef] - Sayad, D.; Zebiri, C.; Daoudi, S.; Benabdelaziz, F. Analysis of the effect of a gyrotropic anisotropy on the phase constant and characteristic impedance of a shielded microstrip line. Adv. Electromagn.
**2019**, 8, 15–22. [Google Scholar] [CrossRef] - Heydari, M.B.; Ahmadvand, A. A novel analytical model for a circularly-polarized; ferrite-based slot antenna by solving an integral equation for the electric field on the circular slot. Waves Random Complex Media
**2020**, 1–20. [Google Scholar] [CrossRef] - Lee, W.; Hong, Y.K.; Choi, M.; Won, H.; Lee, J.; Park, S.O.; Yoon, H.S. Ferrite-cored patch antenna with suppressed harmonic radiation. IEEE Trans. Antennas Propag.
**2018**, 66, 3154–3159. [Google Scholar] [CrossRef] - Kamra, V.; Dreher, A. Efficient analysis of multiple microstrip transmission lines with anisotropic substrates. IEEE Microw. Wirel. Compon. Lett.
**2018**, 28, 636–638. [Google Scholar] [CrossRef] - Buzov, A.L.; Buzova, M.A.; Klyuev, D.S.; Mishin, D.V.; Neshcheret, A.M. Calculating the Input Impedance of a Microstrip Antenna with a Substrate of a Chiral Metamaterial. J. Commun. Technol. Electron.
**2018**, 63, 1259–1264. [Google Scholar] [CrossRef] - Klyuev, D.S.; Minkin, M.A.; Mishin, D.V.; Neshcheret, A.M.; Tabakov, D.P. Characteristics of Radiation from a Microstrip Antenna on a Substrate Made of a Chiral Metamaterial. Radiophys. Quantum Electron.
**2018**, 61, 445–455. [Google Scholar] [CrossRef] - Hu, Y.; Fang, Y.; Wang, D.; Zhan, Q.; Zhang, R.; Liu, Q.H. The scattering of electromagnetic fields from anisotropic objects embedded in anisotropic multilayers. IEEE Trans. Antennas Propag.
**2019**, 67, 7561–7568. [Google Scholar] [CrossRef] - Zebiri, C.; Daoudi, S.; Benabdelaziz, F.; Lashab, M.; Sayad, D.; Ali, N.T.; Abd-Alhameed, R.A. Gyro-chirality effect of bianisotropic substrate on the operational of rectangular microstrip patch antenna. Int. J. Appl. Electromagn. Mech.
**2016**, 51, 249–260. [Google Scholar] [CrossRef] - Zebiri, C.; Benabdelaziz, F.; Sayad, D. Surface waves investigation of a bianisotropic chiral substrate resonator. Prog. Electromagn. Res.
**2012**, 40, 399–414. [Google Scholar] [CrossRef][Green Version] - Eroglu, A.; Lee, J.K. Far field radiation from an arbitrarily oriented Hertzian dipole in the presence of a layered anisotropic medium. IEEE Trans. Antennas Propag.
**2005**, 53, 3963–3973. [Google Scholar] [CrossRef] - Sayad, D.; Benabdelaziz, F.; Zebiri, C.; Daoudi, S.; Abd-Alhameed, R.A. Spectral domain analysis of gyrotropic anisotropy chiral effect on the input impedance of a printed dipole antenna. Prog. Electromagn. Res.
**2016**, 51, 1–8. [Google Scholar] [CrossRef][Green Version] - Braaten, B.D.; Rogers, D.A.; Nelson, R.M. Multi-conductor spectral domain analysis of the mutual coupling between printed dipoles embedded in stratified uniaxial anisotropic dielectrics. IEEE Trans. Antennas Propag.
**2012**, 60, 1886–1898. [Google Scholar] [CrossRef] - Soares, A.; Fonseca, S.B.D.A.; Giarola, A. The effect of a dielectric cover on the current distribution and input impedance of printed dipoles. IEEE Trans. Antennas Propag.
**1984**, 32, 1149–1153. [Google Scholar] [CrossRef] - Nelson, R.M.; Rogers, D.A.; D’Assuncao, A.G. Resonant frequency of a rectangular microstrip patch on several uniaxial substrates. IEEE Trans. Antennas Propag.
**1990**, 38, 973–981. [Google Scholar] [CrossRef] - Davidson, D.B.; Aberle, J.T. An introduction to spectral domain method-of-moments formulations. IEEE Antennas Propag. Mag.
**2004**, 46, 11–19. [Google Scholar] [CrossRef] - Wait, J. Fields of a horizontal dipole over a stratified anisotropic half-space. IEEE Trans. Antennas Propag.
**1966**, 14, 790–792. [Google Scholar] [CrossRef] - Kong, J.A. Electromagnetic fields due to dipole antennas over stratified anisotropic media. Geophysics
**1972**, 37, 985–996. [Google Scholar] [CrossRef][Green Version] - Tang, C.M. Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media. IEEE Trans. Antennas Propag.
**1979**, 27, 665–670. [Google Scholar] [CrossRef] - Lee, J.K.; Kong, J.A. Dyadic Green’s functions for layered anisotropic medium. Electromagnetics
**1983**, 3, 111–130. [Google Scholar] [CrossRef] - Braaten, B.D.; Nelson, R.M.; Rogers, D.A. Input impedance and resonant frequency of a printed dipole with arbitrary length embedded in stratified uniaxial anisotropic dielectrics. IEEE Antennas Wirel. Propag. Lett.
**2009**, 8, 806–810. [Google Scholar] [CrossRef] - Eroglu, A.; Lee, Y.H.; Lee, J.K. Dyadic Green’s functions for multi-layered uniaxially anisotropic media with arbitrarily oriented optic axes. IET Microw. Antennas Propag.
**2011**, 5, 1779–1788. [Google Scholar] [CrossRef] - Wang, N.; Wang, G.P. Effective medium theory with closed-form expressions for bi-anisotropic optical metamaterials. Opt. Express
**2019**, 27, 23739–23750. [Google Scholar] [CrossRef] - Erturk, V.B.; Rojas, R.G. Efficient analysis of input impedance and mutual coupling of microstrip antennas mounted on large coated cylinders. IEEE Trans. Antennas Propag.
**2003**, 51, 739–749. [Google Scholar] [CrossRef] - leukenov, S.K.; Assilbekova, A.M. Surface of wave vectors of electromagnetic waves in anisotropic dielectric media with rhombic symmetry. Telecommun. Radio Eng.
**2017**, 76, 1231–1238. [Google Scholar] [CrossRef] - Sayad, D.; Zebiri, C.; Elfergani, I.; Rodriguez, J.; Abobaker, H.; Ullah, A.; Benabdelaziz, F. Complex bianisotropy effect on the propagation constant of a shielded multilayered coplanar waveguide using improved full generalized exponential matrix technique. Electronics
**2020**, 9, 243. [Google Scholar] [CrossRef][Green Version] - Sayad, D.; Zebiri, C.; Elfergani, I.; Rodriguez, J.; Abd-Alhameed, R.A.; Benabdelaziz, F. Analysis of Chiral and Achiral Medium Based Coplanar Waveguide Using Improved Full Generalized Exponential Matrix Technique. Radioengineering
**2020**, 29, 591–600. [Google Scholar] [CrossRef] - Zebiri, C.; Sayad, D. Effect of bianisotropy on the characteristic impedance of a shielded microstrip line for wideband impedance matching applications. Waves Random Complex Media
**2020**, 1–14. [Google Scholar] [CrossRef] - Nakano, H.; Kerner, S.R.; Alexopoulos, N.G. The moment method solution for printed wire antennas of arbitrary configuration. IEEE Trans. Antennas Propag.
**1988**, 36, 1667–1674. [Google Scholar] [CrossRef][Green Version] - Lee, H.; Tripathi, V.K. Spectral domain analysis of frequency dependent propagation characteristics of planar structures on uniaxial medium. IEEE Trans. Microw. Theory Tech.
**1982**, 30, 1188–1193. [Google Scholar] - Harrington, R.F. Field Computation by Moment Methods; IEEE Press, Inc.: New York, NY, USA, 1992. [Google Scholar]
- Itoh, T. Numerical Techniques for Microwave and Millimeter Wave Passive Structures, 1st ed.; John Wiley and Sons: Hoboken, NJ, USA, 1988. [Google Scholar]
- Bianconi, G.; Mittra, R. Efficient Numerical Techniques for Analyzing Microstrip Circuits and Antennas Etched on Layered Media via the Characteristic Basis Function Method. In Computational Electromagnetics; Springer: New York, NY, USA, 2014; pp. 111–148. [Google Scholar]
- Rana, I.; Alexopoulos, N. Current distribution and input impedance of printed dipoles. IEEE Trans. Antennas Propag.
**1981**, 29, 99–105. [Google Scholar] [CrossRef] - Braaten, B.D.; Rogers, D.A.; Nelson, R.M. Current distribution of a printed dipole with arbitrary length embedded in layered uniaxial anisotropic dielectrics. In Proceedings of the 2009 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC), Belem, Brazil, 3–6 November 2009; pp. 72–77. [Google Scholar]
- MATLAB, version 2018a; The MathWorks, Inc.: Natick, MA, USA, 2018.
- Codreanu, I.; Boreman, G.D. Influence of dielectric substrate on the responsivity of microstrip dipole-antenna-coupled infrared microbolometers. Appl. Opt.
**2002**, 41, 1835–1840. [Google Scholar] [CrossRef]

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

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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